Class 10th Mathematics CBSE Solution
Exercise 4.1- Check whether the following are quadratic equations : (i) (x+1)^2 = 2 (x-3) (ii)…
- Represent the following situations in the form of quadratic equations : (i) The…
Exercise 4.2- Find the roots of the following quadratic equations by factorisation: (i) x^2 -…
- Solve the problems given in Example 1.
- Find two numbers whose sum is 27 and product is 182.
- Find two consecutive positive integers, sum of whose squares is 365.…
- The altitude of a right triangle is 7 cm less than its base. If the hypotenuse…
- A cottage industry produces a certain number of pottery articles in a day. It…
Exercise 4.3- Find the roots of the following quadratic equations, if they exist, by the…
- Find the roots of the quadratic equations given in Q.1 above by applying the…
- Find the roots of the following equations: (i) x - 1/x = 3 , x not equal 0 (ii)…
- The sum of the reciprocals of Rehmans ages, (in years) 3 years ago and 5 years…
- In a class test, the sum of Shefalis marks in Mathematics and English is 30. Had…
- The diagonal of a rectangular field is 60 metres more than the shorter side. If…
- The difference of squares of two numbers is 180. The square of the smaller…
- A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it…
- Two water taps together can fill a tank in 9 3/8 hours. The tap of larger…
- An express train takes 1 hour less than a passenger train to travel 132 km…
- Sum of the areas of two squares is 468 m^2 . If the difference of their…
Exercise 4.4- Find the nature of the roots of the following quadratic equations. If the real…
- Find the values of k for each of the following quadratic equations, so that they…
- Is it possible to design a rectangular mango grove whose length is twice its…
- Is the following situation possible? If so, determine their present ages. The…
- Is it possible to design a rectangular park of perimeter 80 m and area 400 m^2 ?…
- Check whether the following are quadratic equations : (i) (x+1)^2 = 2 (x-3) (ii)…
- Represent the following situations in the form of quadratic equations : (i) The…
- Find the roots of the following quadratic equations by factorisation: (i) x^2 -…
- Solve the problems given in Example 1.
- Find two numbers whose sum is 27 and product is 182.
- Find two consecutive positive integers, sum of whose squares is 365.…
- The altitude of a right triangle is 7 cm less than its base. If the hypotenuse…
- A cottage industry produces a certain number of pottery articles in a day. It…
- Find the roots of the following quadratic equations, if they exist, by the…
- Find the roots of the quadratic equations given in Q.1 above by applying the…
- Find the roots of the following equations: (i) x - 1/x = 3 , x not equal 0 (ii)…
- The sum of the reciprocals of Rehmans ages, (in years) 3 years ago and 5 years…
- In a class test, the sum of Shefalis marks in Mathematics and English is 30. Had…
- The diagonal of a rectangular field is 60 metres more than the shorter side. If…
- The difference of squares of two numbers is 180. The square of the smaller…
- A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it…
- Two water taps together can fill a tank in 9 3/8 hours. The tap of larger…
- An express train takes 1 hour less than a passenger train to travel 132 km…
- Sum of the areas of two squares is 468 m^2 . If the difference of their…
- Find the nature of the roots of the following quadratic equations. If the real…
- Find the values of k for each of the following quadratic equations, so that they…
- Is it possible to design a rectangular mango grove whose length is twice its…
- Is the following situation possible? If so, determine their present ages. The…
- Is it possible to design a rectangular park of perimeter 80 m and area 400 m^2 ?…
Exercise 4.1
Question 1.Check whether the following are quadratic equations :
(i) (x + 1)2 = 2 (x - 3)
(ii) x2 – 2x = (-2)(3 – x )
(iii) (x – 2 )(x + 1) = (x – 1)(x + 3)
(iv) (x – 3 )(2x + 1) = x(x + 5)
(v) (2x – 1)(x – 3 ) = (x + 5)(x – 1 )
(vi) x2 + 3x + 1 = (x – 2 )2
(vii) (x + 2)3 = 2x (x2 - 1)
(viii) x3 – 4x2 – x + 1 = (x - 2)3
Answer:For a equation to be quadratic equation, degree of the equation(highest power of the variable in the equation) should be 2
Thus a quadratic equation is of the form,
ax2 + bx + c = 0 , where a ≠ 0
(i) (x + 1)2 = 2 (x - 3)
We know (a + b)2=a2 + b2 + 2 ab
⇒ x2 + 2x + 1 = 2x - 6
⇒ x2 + 2x + 1 - 2x + 6 = 0
⇒ x2 + 7 = 0
Degree (highest power) of the equation is 2
Equation is in the form of ax2+bx+c = 0, with a ≠ 0
So, we can say that the given equation is a quadratic equation.
(ii) x2 – 2x = (-2)(3 – x )
⇒ x2 – 2x = -6 + 2x
⇒ x2 – 2x +6 - 2x=0
⇒ x2 - 4x + 6 = 0
Equation is of the form ax2+bx+c = 0 with a ≠ 0
Hence, the given equation is a quadratic equation.
(iii) (x – 2 )(x + 1) = (x – 1)(x + 3)
x(x + 1) - 2(x + 1) = x(x + 3) -1 (x + 3)
⇒ x2 + x - 2x - 2 = x2 + 3x - x - 3
⇒ x2 – x – 2 = x2 + 2x – 3
⇒ x2 - x - 2 - x2 - 2x + 3 = 0
⇒ - 3x + 1 = 0
⇒ 3x – 1 = 0
Degree (highest power) of the equation is 1
Equation is not in the form ax2 + bx + c = 0.
Hence, the given equation is not a quadratic equation.
(iv) (x – 3 )(2x + 1) = x(x + 5)
⇒ x(2x + 1)-3(2x + 1) = x2 + 5x
⇒ 2x2 + x - 6x - 3 = x2 + 5x
⇒ 2x2 – 5x – 3 = x2 + 5x
⇒ 2x2 – 5x – 3 - x2 - 5x = 0
⇒ x2 – 10x – 3 = 0
Degree of the equation is 2
Equation is in the form ax2 + bx + c = 0, with a ≠ 0
Hence, the given equation is a quadratic equation.
(v) (2x – 1)(x – 3 ) = (x + 5)(x – 1 )
⇒ 2x(x - 3) -1(x - 3) = x(x - 1) + 5(x - 1)
⇒ 2x2 - 6x - x + 3 = x2 - x + 5x - 5
⇒ 2x2 – 7x + 3 = x2 + 4x – 5
⇒ 2x2 – 7x + 3 - x2 - 4x + 5 = 0
⇒ x2 – 11x + 8 = 0
Degree of the equation is 2
Equation is in the form ax2+bx+c = 0, with a ≠ 0
Hence, the given equation is a quadratic equation.
(vi) x2 + 3x + 1 = (x – 2 )2
⇒ x2 + 3x + 1 = x2 + 4 – 4x
⇒ x2 + 3x + 1 - x2 - 4 + 4x = 0
⇒ 7x – 3 = 0
Degree of the equation is 1
It is not of the form ax2 + bx + c = 0.
Hence, the given equation is not a quadratic equation.
(vii) (x + 2)3 = 2x (x2 - 1)
Apply the formula (a+b)3 = a3 + b3 + 3ab(a + b)
⇒ x3 + 8 + 6x2 + 12x = 2x3 – 2x
⇒ x3 + 8 + 6x2 + 12x - 2x3 + 2x = 0
⇒ - x3 + 8 + 6x2 + 14x= 0
⇒ x3 – 14x - 6x2 – 8 = 0
Degree of the equation is 3
It is not of the form ax2 + bx + c = 0.
Hence, the given equation is not a quadratic equation.
(viii) x3 – 4x2 – x + 1 = (x - 2)3
(a - b)3 = a3 - b3 - 3 a2b + 3ab2
⇒ x3 – 4x2 – x +1 = x3 – 8 – 6x2 +12x
⇒ x3 – 4x2 – x +1 - x3 + 8 + 6x2 -12x = 0
⇒ 2x2 – 13x + 9 = 0
Degree of the equation is 2 and
It is of the form ax2+bx+c = 0, with a ≠ 0
Hence, the given equation is a quadratic equation.
Question 2.Represent the following situations in the form of quadratic equations :
(i) The area of a rectangular plot is 528 m2. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.
(ii) The product of two consecutive positive integers is 306. We need to find the integers.
(iii) Rohan’s mother is 26 years older than him. The product of their ages (in years)3 years from now will be 360. We would like to find Rohan’s present age.
(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.
Answer:(i)The area of a rectangular plot is 528 m2.
Let the breadth of the plot be x m.
The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.
Thus, the length of the plot is (2x + 1) m.
Area of a rectangle = Length × Breadth
∴ 528 = x (2x + 1)
⇒ 2x2 + x – 528 = 0 (required quadratic form)
(ii)The product of two consecutive positive integers is 306. We need to find the integers.
Let the consecutive integers be x and x + 1.
It is given that their product is 306.
∴ x (x+1) = 306
⇒ x2 + x – 306 = 0 (required quadratic form)
(iii)
Let Rohan’s present age be x.
Given, Rohan's Mother is 26 years older than him
Hence, his mother’s age = x+26
3 years hence,
Rohan’s age = x + 3
Mother’s age = x + 26 + 3
= x + 29
It is given that the product of their ages after 3 years is 360.
∴ (x+3) (x+29) = 360
x2 + 3x + 29x + 87 = 360
⇒ x2 +32x – 273 = 0 (required quadratic form)
(iv)
Let the speed of train be x km/h.
As speed = distance / time
⇒ Time taken for travel 480 km = ![](data:image/png;base64,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)
In the second condition,
speed of train = (x – 8) km/h
Given that the train will take 3 hours more to cover the same distance.
Therefore, Time taken for traveling 480 km = ![](data:image/png;base64,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)
Speed × Time = Distance
= 480
⇒ 480 + 3x -
- 24 = 480
⇒ 3x -
= 24
⇒ 3x2 - 3840 = 24x
⇒ 3x2 – 24x - 3840 = 0
⇒ x2 – 8x - 1280 = 0 (required quadratic form)
Check whether the following are quadratic equations :
(i) (x + 1)2 = 2 (x - 3)
(ii) x2 – 2x = (-2)(3 – x )
(iii) (x – 2 )(x + 1) = (x – 1)(x + 3)
(iv) (x – 3 )(2x + 1) = x(x + 5)
(v) (2x – 1)(x – 3 ) = (x + 5)(x – 1 )
(vi) x2 + 3x + 1 = (x – 2 )2
(vii) (x + 2)3 = 2x (x2 - 1)
(viii) x3 – 4x2 – x + 1 = (x - 2)3
Answer:
For a equation to be quadratic equation, degree of the equation(highest power of the variable in the equation) should be 2
Thus a quadratic equation is of the form,
ax2 + bx + c = 0 , where a ≠ 0
(i) (x + 1)2 = 2 (x - 3)
We know (a + b)2=a2 + b2 + 2 ab
⇒ x2 + 2x + 1 = 2x - 6
⇒ x2 + 2x + 1 - 2x + 6 = 0
⇒ x2 + 7 = 0
Equation is in the form of ax2+bx+c = 0, with a ≠ 0
So, we can say that the given equation is a quadratic equation.
(ii) x2 – 2x = (-2)(3 – x )
⇒ x2 – 2x = -6 + 2x
⇒ x2 – 2x +6 - 2x=0
⇒ x2 - 4x + 6 = 0
Equation is of the form ax2+bx+c = 0 with a ≠ 0
Hence, the given equation is a quadratic equation.
(iii) (x – 2 )(x + 1) = (x – 1)(x + 3)
x(x + 1) - 2(x + 1) = x(x + 3) -1 (x + 3)⇒ x2 + x - 2x - 2 = x2 + 3x - x - 3
⇒ x2 – x – 2 = x2 + 2x – 3
⇒ x2 - x - 2 - x2 - 2x + 3 = 0⇒ - 3x + 1 = 0
⇒ 3x – 1 = 0
Degree (highest power) of the equation is 1Equation is not in the form ax2 + bx + c = 0.
Hence, the given equation is not a quadratic equation.
(iv) (x – 3 )(2x + 1) = x(x + 5)
⇒ x(2x + 1)-3(2x + 1) = x2 + 5x⇒ 2x2 + x - 6x - 3 = x2 + 5x
⇒ 2x2 – 5x – 3 = x2 + 5x
⇒ 2x2 – 5x – 3 - x2 - 5x = 0
⇒ x2 – 10x – 3 = 0
Degree of the equation is 2
Equation is in the form ax2 + bx + c = 0, with a ≠ 0
Hence, the given equation is a quadratic equation.
(v) (2x – 1)(x – 3 ) = (x + 5)(x – 1 )
⇒ 2x(x - 3) -1(x - 3) = x(x - 1) + 5(x - 1)⇒ 2x2 - 6x - x + 3 = x2 - x + 5x - 5
⇒ 2x2 – 7x + 3 = x2 + 4x – 5
⇒ 2x2 – 7x + 3 - x2 - 4x + 5 = 0
⇒ x2 – 11x + 8 = 0
Degree of the equation is 2Equation is in the form ax2+bx+c = 0, with a ≠ 0
Hence, the given equation is a quadratic equation.
(vi) x2 + 3x + 1 = (x – 2 )2
⇒ x2 + 3x + 1 = x2 + 4 – 4x
⇒ x2 + 3x + 1 - x2 - 4 + 4x = 0
⇒ 7x – 3 = 0
Degree of the equation is 1It is not of the form ax2 + bx + c = 0.
Hence, the given equation is not a quadratic equation.
(vii) (x + 2)3 = 2x (x2 - 1)
Apply the formula (a+b)3 = a3 + b3 + 3ab(a + b)
⇒ x3 + 8 + 6x2 + 12x = 2x3 – 2x
⇒ x3 + 8 + 6x2 + 12x - 2x3 + 2x = 0⇒ - x3 + 8 + 6x2 + 14x= 0
⇒ x3 – 14x - 6x2 – 8 = 0
Degree of the equation is 3It is not of the form ax2 + bx + c = 0.
Hence, the given equation is not a quadratic equation.
(viii) x3 – 4x2 – x + 1 = (x - 2)3
(a - b)3 = a3 - b3 - 3 a2b + 3ab2⇒ x3 – 4x2 – x +1 = x3 – 8 – 6x2 +12x
⇒ x3 – 4x2 – x +1 - x3 + 8 + 6x2 -12x = 0⇒ 2x2 – 13x + 9 = 0
Degree of the equation is 2 andIt is of the form ax2+bx+c = 0, with a ≠ 0
Hence, the given equation is a quadratic equation.
Question 2.
Represent the following situations in the form of quadratic equations :
(i) The area of a rectangular plot is 528 m2. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.
(ii) The product of two consecutive positive integers is 306. We need to find the integers.
(iii) Rohan’s mother is 26 years older than him. The product of their ages (in years)3 years from now will be 360. We would like to find Rohan’s present age.
(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.
Answer:
(i)The area of a rectangular plot is 528 m2.
Let the breadth of the plot be x m.
The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.
Thus, the length of the plot is (2x + 1) m.
Area of a rectangle = Length × Breadth
∴ 528 = x (2x + 1)
⇒ 2x2 + x – 528 = 0 (required quadratic form)
(ii)The product of two consecutive positive integers is 306. We need to find the integers.
Let the consecutive integers be x and x + 1.
It is given that their product is 306.
∴ x (x+1) = 306
⇒ x2 + x – 306 = 0 (required quadratic form)
(iii)
Let Rohan’s present age be x.
Given, Rohan's Mother is 26 years older than him
Hence, his mother’s age = x+26
3 years hence,
Rohan’s age = x + 3
Mother’s age = x + 26 + 3
= x + 29
It is given that the product of their ages after 3 years is 360.
∴ (x+3) (x+29) = 360
x2 + 3x + 29x + 87 = 360⇒ x2 +32x – 273 = 0 (required quadratic form)
(iv)
Let the speed of train be x km/h.
As speed = distance / time
⇒ Time taken for travel 480 km =
In the second condition,
speed of train = (x – 8) km/h
Given that the train will take 3 hours more to cover the same distance.
Therefore, Time taken for traveling 480 km =
Speed × Time = Distance
= 480
⇒ 480 + 3x - - 24 = 480
⇒ 3x - = 24
⇒ 3x2 – 24x - 3840 = 0
⇒ x2 – 8x - 1280 = 0 (required quadratic form)
Exercise 4.2
Question 1.Find the roots of the following quadratic equations by factorisation:
(i) ![](data:image/png;base64,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)
(ii) ![](data:image/png;base64,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)
(iii) ![](data:image/png;base64,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)
(iv) ![](data:image/png;base64,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)
(v) ![](data:image/png;base64,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)
Answer:(i) X2 – 3x – 10
= x2 – 5x +2x – 10
= x(x – 5 ) + 2(x – 5 )
= (x – 5 )(x+2)
Roots of this equation are the values for which (x – 5 )(x+2) = 0
∴ x - 5 = 0 or x + 2 = 0
i.e.,
x = 5 or x = -2
(ii) 2 x2 + x – 6
= 2x2 + 4x – 3x – 6
= 2x (x+2) – 3(x+2)
= (x+2)(2x – 3 )
Roots of this equation are the values for which (x+2)(2x – 3) = 0
∴ x+ 2= 0 or 2x – 3 = 0
x = −2 or x = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAiCAMAAACk7TNkAAAAAXNSR0IArs4c6QAAADlQTFRFAAAAAAAAAAA6AGa2OgAAOgA6OpDbZgAAZgA6Zrb/kDoAkDo6kNv/tmYA25A6/7Zm/9uQ//+2///b3Fh/VQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAXElEQVQoU42OSxKAIAxDE/8ognD/wxrqhxk3mM2btmkboKVAcgLy5hE7b+5jNObVFeykUf3BIS3eiHuvdf6Z60vRX3v1pZnsYX8/+RCUW4oaS+FCqaJyau0NXq+dL+oCP6NdZREAAAAASUVORK5CYII=)
(iii) ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Roots of this equation are the values for which
= 0
∴
= 0 or
= 0
![](data:image/png;base64,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)
(iv) ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHIAAAAsCAMAAACOhxzIAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6Oma2OpDbZgAAZgA6ZjqQZmZmZrb/kDoAkDpmkLaQkNv/tmYAtmY6tv//25A627Zm2////7Zm/9uQ//+2///bUQWovgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABo0lEQVRYR+1WXXOEIAw87q62Zz9stdKL8P9/ZxMkiPYEoXPMdGqefAB2N9kkHg57/IMMDI9tWZWqFseykHDqZWFITOkOeSdf7YndE5ubAd0Iiofc+7fu4UBLflCe+gwK3flqbqnn9gBpE003lwxAvDJUb3wxbVV4iFs7lgEmIJnEGge95TpUm9Lj7SJpcwtJ3tANH1f16xZIfxdZmTIJ0StIdxlNgKbGNHUBeFeAsSikEVxZ475wXoPzdfz+bNHBMvTEVPMO0ahHhKDznWlAimC6GNK3Ohw/nkJkPUhbzMBpR0PwIduTJkUIb9SpemrUW8yTIH+ysSqByRBkrFPXIBMTS2Q4ySDC+ZrXMu6XxQlvhDCkevkKT8CFY1Mxp77kHzH93iMPCMwTB5k25xy1afpgPdHc9B625mp3+7uIp09Y51AtdlfMK4HntomU1LMw05CN6dtgnZhujBe7uT1+ty8j7mHI+NBIteH6eaCRDWYiFgsawln/HdkMhwpVSpJaKsZa6qagTlWbFl849q6CWWVJx5oyGtuWC1qMW/6qyjH6W0jfGIUXG3C/OZ4AAAAASUVORK5CYII=)
Roots of this equation are the values for which (4x – 1 )2 = 0
Therefore,
(4x – 1 ) = 0 or (4x – 1) = 0
i.e., ![](data:image/png;base64,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)
(v) 100x2 – 20x +1
= 100x2 – 10x – 10x +1
=10x(10x – 1) – 1 (10x – 1)
=(10x – 1)2
Roots of this equation are the values for which (10x – 1)2 = 0
Therefore,
(10x – 1) = 0 or (10x – 1) = 0
i.e., ![](data:image/png;base64,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)
Question 2.Solve the problems given in Example 1.
Answer:(i) Let the number of John’s marbles be x.
Therefore, number of Jivanti’s marble = 45 − x
After losing 5 marbles,
Number of John’s marbles = x − 5
Number of Jivanti’s marbles = 45 − x − 5 = 40 − x
Given that the product of their marbles is 124.
∴ (x – 5) (40 – x) = 124
⇒ x2 – 45x +324 = 0
Now, to factorize this equation, we need to take numbers such that their product is 324 and sum is 45
⇒ x2 – 36x – 9x +324 = 0
⇒ x(x – 36 ) – 9 (x – 36 ) = 0
⇒ (x – 36) (x – 9) = 0
Either x – 36 = 0 or x − 9 = 0
i.e.,
x = 36 or x = 9
If the number of John’s marbles = 36,
Then, number of Jivanti’s marbles = 45 − 36 = 9
If number of John’s marbles = 9,
Then, number of Jivanti’s marbles = 45 − 9 = 36
(ii) Let the number of toys produced be x.
∴ Cost of production of each toy = Rs (55 − x)
It is given that, total production of the toys = Rs 750
∴ x(55 – x) = 750
⇒ x2 – 55x + 750 = 0
Now to factorize this equation we have to find two numbers such that their product is 750 and sum is 55
⇒ x2 – 25x – 30x + 750 = 0
⇒ x(x – 25 ) – 30(x – 25 ) = 0
⇒ (x – 25)(x – 30) = 0
Either x – 25 = 0 or x − 30 = 0
i.e., x = 25 or x = 30
Hence, the number of toys will be either 25 or 30.
Question 3.Find two numbers whose sum is 27 and product is 182.
Answer:Let the first number be x and the second number is 27 − x.
[As the sum of both the numbers is 27]
Therefore, their product = x (27 − x)
It is given that the product of these numbers is 182.
x (27 – x) = 182
- x2 + 27x - 182 = 0
Changing the signs on both sides we get,
x2 - 27x + 182 = 0
Factorizing we get , 13 and 14 are the numbers whose sum is 27 and product is 182
x2 – 13x – 14x + 182 = 0
= x(x – 13) – 14 (x – 13)= 0
= (x – 13) (x – 14) = 0
Either x – 13 = 0 or x − 14 = 0
i.e., x = 13 or x = 14
If first number = 13, then
Other number = 27 − 13 = 14
If first number = 14, then
Other number = 27 − 14 = 13
Therefore, the numbers are 13 and 14.
Question 4.Find two consecutive positive integers, sum of whose squares is 365.
Answer:To Find: Consecutive integers sum of whose square is 365
Consecutive integers mean that the difference between the integers is of 1
Let the consecutive positive integers be x and x + 1.
Given that x2 + (x + 1)2 = 365
⇒ x2 + x2 + 1 + 2x = 365
⇒ 2x2 + 2x – 364 = 0
⇒ x2 + x – 182 = 0
Now to factorize the above quadratic equation, we need to chose numbers such that their product is 182 and difference is 1
⇒ x2 + 14x – 13x – 182 = 0
⇒ x (x + 14) – 13 (x + 14) = 0
(x+14)(x – 13) = 0
Either x + 14 = 0 or x − 13 = 0, i.e., x = −14 or x = 13
Since the question ask for positive integers, x can only be 13.
∴ x + 1 = 13 + 1 = 14
Therefore, two consecutive positive integers will be 13 and 14.
Question 5.The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.
Answer:Consider a right angle ΔABC
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Let the base of the Δ be x
⇒ Altitude = x - 7
In the right-angled triangle
According to the Pythagoras theorem
(Hypotenuse)2 = (base)2 + (altitude)2
⇒ 132 = x2 + (x - 7)2
⇒ 169 – x2 – x2 - 49 + 14x = 0
⇒ 2x2 - 14x – 120 = 0
⇒ x2 – 7x - 60 = 0
⇒ x2 – 12x + 5x - 60 = 0
⇒ x( x - 12) + 5( x – 12) = 0
⇒ (x + 5) (x - 12) = 0
⇒ x = - 5 or 12
But base cannot be negative, so x = 12 cm
⇒ altitude = 12 - 7 = 5 cm
Hence 5 cm and 12 cm are the two sides of the given triangle.
Question 6.A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article.
Answer:To find: Number of articles produced and cost of each article.
Let the number of articles produced be x.
Therefore, cost of production of each article = Rs (2x + 3)
The total cost of production = Total quantity produced x cost of one article
It is given that the total cost of production is Rs 90.
∴ x (2x + 3) = 90
⇒ 2x2 + 3x = 90
⇒ 2x2 + 3x – 90 = 0
⇒ 2x2 +15x - 12x-90 =0
⇒ x (2x + 15) - 6(2x + 15) = 0
⇒(2x + 15) (x - 6) = 0
Either 2x + 15 = 0 or x − 6 = 0,
i.e., x =
or x = 6
As the number of articles produced can only be a positive integer,
Therefore, x can only be 6.
Hence, the number of articles produced = 6
Cost of each article = 2 × 6 + 3
= 12 + 3
= Rs 15
Find the roots of the following quadratic equations by factorisation:
(i)
(ii)
(iii)
(iv)
(v)
Answer:
(i) X2 – 3x – 10
= x2 – 5x +2x – 10
= x(x – 5 ) + 2(x – 5 )
= (x – 5 )(x+2)
Roots of this equation are the values for which (x – 5 )(x+2) = 0
∴ x - 5 = 0 or x + 2 = 0
i.e.,
x = 5 or x = -2
(ii) 2 x2 + x – 6
= 2x2 + 4x – 3x – 6
= 2x (x+2) – 3(x+2)
= (x+2)(2x – 3 )
Roots of this equation are the values for which (x+2)(2x – 3) = 0
∴ x+ 2= 0 or 2x – 3 = 0
x = −2 or x =
(iii)
Roots of this equation are the values for which = 0
∴ = 0 or
= 0
(iv)
Roots of this equation are the values for which (4x – 1 )2 = 0
Therefore,
(4x – 1 ) = 0 or (4x – 1) = 0
i.e.,
(v) 100x2 – 20x +1
= 100x2 – 10x – 10x +1
=10x(10x – 1) – 1 (10x – 1)
=(10x – 1)2
Roots of this equation are the values for which (10x – 1)2 = 0
Therefore,
(10x – 1) = 0 or (10x – 1) = 0
i.e.,
Question 2.
Solve the problems given in Example 1.
Answer:
(i) Let the number of John’s marbles be x.
Therefore, number of Jivanti’s marble = 45 − x
After losing 5 marbles,
Number of John’s marbles = x − 5
Number of Jivanti’s marbles = 45 − x − 5 = 40 − x
Given that the product of their marbles is 124.
∴ (x – 5) (40 – x) = 124
⇒ x2 – 45x +324 = 0
Now, to factorize this equation, we need to take numbers such that their product is 324 and sum is 45
⇒ x2 – 36x – 9x +324 = 0
⇒ x(x – 36 ) – 9 (x – 36 ) = 0
⇒ (x – 36) (x – 9) = 0
Either x – 36 = 0 or x − 9 = 0
i.e.,
x = 36 or x = 9
If the number of John’s marbles = 36,
Then, number of Jivanti’s marbles = 45 − 36 = 9
If number of John’s marbles = 9,
Then, number of Jivanti’s marbles = 45 − 9 = 36
(ii) Let the number of toys produced be x.
∴ Cost of production of each toy = Rs (55 − x)
It is given that, total production of the toys = Rs 750
∴ x(55 – x) = 750
⇒ x2 – 55x + 750 = 0
Now to factorize this equation we have to find two numbers such that their product is 750 and sum is 55
⇒ x2 – 25x – 30x + 750 = 0
⇒ x(x – 25 ) – 30(x – 25 ) = 0
⇒ (x – 25)(x – 30) = 0
Either x – 25 = 0 or x − 30 = 0
i.e., x = 25 or x = 30
Hence, the number of toys will be either 25 or 30.
Question 3.
Find two numbers whose sum is 27 and product is 182.
Answer:
Let the first number be x and the second number is 27 − x.
[As the sum of both the numbers is 27]
Therefore, their product = x (27 − x)
It is given that the product of these numbers is 182.
x (27 – x) = 182
- x2 + 27x - 182 = 0
Changing the signs on both sides we get,
x2 - 27x + 182 = 0
Factorizing we get , 13 and 14 are the numbers whose sum is 27 and product is 182
x2 – 13x – 14x + 182 = 0
= x(x – 13) – 14 (x – 13)= 0
= (x – 13) (x – 14) = 0
Either x – 13 = 0 or x − 14 = 0
i.e., x = 13 or x = 14
If first number = 13, then
Other number = 27 − 13 = 14
If first number = 14, then
Other number = 27 − 14 = 13
Therefore, the numbers are 13 and 14.
Question 4.
Find two consecutive positive integers, sum of whose squares is 365.
Answer:
To Find: Consecutive integers sum of whose square is 365
Consecutive integers mean that the difference between the integers is of 1
Let the consecutive positive integers be x and x + 1.
Given that x2 + (x + 1)2 = 365
⇒ x2 + x2 + 1 + 2x = 365
⇒ 2x2 + 2x – 364 = 0
⇒ x2 + x – 182 = 0
Now to factorize the above quadratic equation, we need to chose numbers such that their product is 182 and difference is 1
⇒ x2 + 14x – 13x – 182 = 0
⇒ x (x + 14) – 13 (x + 14) = 0
(x+14)(x – 13) = 0
Either x + 14 = 0 or x − 13 = 0, i.e., x = −14 or x = 13
Since the question ask for positive integers, x can only be 13.
∴ x + 1 = 13 + 1 = 14
Therefore, two consecutive positive integers will be 13 and 14.
Question 5.
The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.
Answer:
Consider a right angle ΔABC
Let the base of the Δ be x
⇒ Altitude = x - 7
In the right-angled triangle
According to the Pythagoras theorem
(Hypotenuse)2 = (base)2 + (altitude)2
⇒ 132 = x2 + (x - 7)2
⇒ 169 – x2 – x2 - 49 + 14x = 0
⇒ 2x2 - 14x – 120 = 0
⇒ x2 – 7x - 60 = 0
⇒ x2 – 12x + 5x - 60 = 0
⇒ x( x - 12) + 5( x – 12) = 0
⇒ (x + 5) (x - 12) = 0
⇒ x = - 5 or 12
But base cannot be negative, so x = 12 cm
⇒ altitude = 12 - 7 = 5 cm
Hence 5 cm and 12 cm are the two sides of the given triangle.
Question 6.
A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article.
Answer:
To find: Number of articles produced and cost of each article.
Let the number of articles produced be x.
Therefore, cost of production of each article = Rs (2x + 3)
The total cost of production = Total quantity produced x cost of one articleIt is given that the total cost of production is Rs 90.
∴ x (2x + 3) = 90
⇒ 2x2 + 3x = 90⇒ 2x2 + 3x – 90 = 0
⇒ 2x2 +15x - 12x-90 =0
⇒ x (2x + 15) - 6(2x + 15) = 0
⇒(2x + 15) (x - 6) = 0
Either 2x + 15 = 0 or x − 6 = 0,
i.e., x = or x = 6
As the number of articles produced can only be a positive integer,
Therefore, x can only be 6.
Hence, the number of articles produced = 6
Cost of each article = 2 × 6 + 3
= 12 + 3
= Rs 15
Exercise 4.3
Question 1.Find the roots of the following quadratic equations, if they exist, by the method of completing the square:
(i) 2x2 - 7x + 3 = 0
(ii) 2x2 + x - 4 = 0
(iii) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKgAAAAeCAYAAAC42FXcAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAMFSURBVHja7ZpBTuMwFIZ9AA4wSHh2HKB15gqdMOICbcQOsUKWRhwg7a6L4QAzO3bMgjUbTtAbzKI34A4gpyTNpHZix8+JgX/xSyBB897/Pr/n2GWr1YpBUKyCCRAAhSAACgFQCAKgEARACYxj7MVF8AyADg4ofACgUXdPePHBAF2vr47SE/a0H338OZX5BIBCUQCa5xfHQmR5+fsyE+fsJH26Wq+PMN6h3oAuBLtTRRHZ8pwUWJlOOE//Dg2obz51QGWW/kg4f6y/EHGePM5lfhpbsWOIVU3RheC3nLHncopysbg1MdD5gdczflMmQwmo6qZTxjdDj3jffOrjvVhgldFNie1Fnh/HAmcMsR5u8WoyTFKrYiZJ8psS0GK0j1BAinyagCY8uZ9LOfmvCAl7KLrT7PomJkDHjnVX9x2MZdfO5fx0ytnGVI/OD1OBlz+bClrsLRnb6lZB0SmLAHYvRMV4FYu7oQvkko/v/rP+rL4+jeHLWFurClwNF+3j4O0fbAqqS7RpumnMuICinuNqZp98+gK67wj6CWHjUygfXGMNM94Pn9W2cK26oU1BmwGE6giuhembj+vxUtkdlL6K739Mm34qn3wAtY01OkCrD2n8oW1Bq06VpA+hxpVLYXzz6QuoylvM5GVIn6gA7YpV/z96tcVDAmi1F+hQW2H3iYhtiOMLl8L0yacNQpv95+4YpfsYy9cnihFvG2s0HdQXULWnKR7Ucmzgt9oNMrx0uebT9QUP2wP65p6XwicfH3xijW4P2lbotlVWmc6nm7Mz/ivUqqToHDb5NGF0ud5sKzqVTxQ+uADqO+JJ3+JdC1o3vdz4m1bEewK0DqQOUDkTl+oWRMrll8M3Y80xE6FPrj64xjrkOWh5eO90Dmpb0GoFvpke+nxtKECbXVTXPeu3Ul03I9Q+ufrgEmv4MU90k9RVUJPpXXuO9wRoCaYOUJVjln77yTn7V38rbt4th/DJ1QfbWIeAlPQu/rML34of2X+YAEAB6AeAFD4AUAgCoBAAhSAyvQJegOLzItx6CQAAAABJRU5ErkJggg==)
(iv) 2x2 + x + 4 = 0
Answer:i) 2x2 – 7x +3 =0
Dividing by coefficient of x2, we get
![](data:image/png;base64,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)
Adding and subtracting the square of ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Use the formula (a-b)2 = a2 + b2 - 2ab to get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
So,
![](data:image/png;base64,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)
When, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
When, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Hence the value of x is 3 and ![](data:image/png;base64,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)
ii) 2x2 + x – 4 = 0
Dividing by coefficient of ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAAZCAMAAADkFeWYAAAAAXNSR0IArs4c6QAAAFRQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6Oma2OpDbZgAAZgA6ZjqQZrb/kDoAkDpmkLaQkNv/tmYAtmY6tv//25A627Zm2////7Zm/9uQ//+2///bQqrffQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAdElEQVQoU7WQzRKAIAiExf6tLEvK9P3fM5jpUFjHODHLsnyDUj9WNADVMz/2TgXtsqNHk2vYZbYg4siAl5QskH/hbHaFkWarw8Ijd4wCwB2N9NzK5GgKL7VkXwCg3IUvDpvcTZM/6jHc15HACDHn//H/H9EnXugEXNvnuwYAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
Adding and subtracting the square of ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Use the formula (a+b)2 = a2 + b2 + 2ab to get,
![](data:image/png;base64,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)
iii)![](data:image/png;base64,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)
Divide the whole equation by 4 to get,
![](data:image/png;base64,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)
Add and subtract the square of half of the coefficient of x,
![](data:image/png;base64,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)
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)
Use the formula (a+b)2 = a2 + b2 + 2ab to get,
![](data:image/png;base64,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)
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)
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CevC9gUGqOWXcootPU1mSy45kbGBCkZCOXf7gZyGnT2moJoHELwwrlLToLkX/cqShPYddNdSpagWxTl7q2NmEJzBEM+A6US0O55Zkek/cw4aUZNTQsPyaV5JblqAG1XUQGU3HXzY1PAEk1nlpVqnY95lPb/xzBwGKhs6qNIPlm6AUtx7rokmeeVpJoVrH8hPkQgCS87LIhPwkBiyAcKjFql/M7ur7nCgYWgh2Kosx3sMy0O7qF2mJCbuI56ep6NfiFGGVq7qopHPP0Uhti35yGXc3p6PudMxjIKUerVelG7PyQyUFHv1G2nOAPJrlTcRoq993aDCQwZzAgXiqvE8glL0qbja1JMIMlm/wQEYwQwyRCDblSb/ITPNYBzh0MrAtnGUBQu3AfHvJj3Qs15qX8OYKRUCbGY2szksAxgAFxu2SF00w9Ade3tbZ/CQhlSo8+r63B/oVf443HAgadhiBFFhg8upkMNbZH7z6kmHNc0giEfVuboQSOCQxmKP425CaB6UiggcF01qKNpEngoBJoYHBQ8beXNwlMRwINDKazFm0kTQIHlUADg4OKv728SWA6EmhgMJ21aCNpEjioBN4PF13fZlBiJnsAAAAASUVORK5CYII=)
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)
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fZdAlIQAKKuDYgAQlIoGICinjFnWfTJSABCSji2oAEJCCBigko4hV3nk2XgAQkoIhrAxKQgAQqJqCIV9x5Nl0CEpCAIq4NSEACEqiYgCJecefZdAlIQAKKuDYgAQlIoGICinjFnWfTJSABCSji2oAEJCCBigko4hV3nk2XgAQkoIhrAxKQgAQqJqCIV9x5Nl0CEpDAfwCUblqVcPNLjwAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
∴ Roots of equation are = x = ![](data:image/png;base64,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)
iv) 2x2 + x + 4
Divide by coefficient of ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFMAAAAZCAMAAAB6r+I4AAAAAXNSR0IArs4c6QAAAG9QTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6OmZmOmaQOma2OpDbZgAAZgA6ZgBmZjqQZra2Zrb/kDoAkDo6kDpmkLaQkNu2kNv/tmYAtmY6ttuQtv//25A625Bm27Zm2/+22////7Zm/9uQ//+2///bTPYHUAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABe0lEQVRIS+1U23aDIBCEJLWXWBPtRVNpEy7//42dXQXkFNuH0D51HzzKwuzszqAQ//FrE7C1lDdl0e1jL/SmLwsKNHNbHlPti9PUV4zTdVk+6gpIoWXre3SdBP5A2hBL7RNKbkes7y5iejNVYgot5ea1wfqcAAxi5nTq1XZUhERWkrHYACTXAdN1VAsJbPRMFD5NRRAhQeVD6M3LQ0YZQjAVAT/1eEHViGkq9GZrPEKCK4ewday/WCarqjt4y6AikdA0pCkYneFCgiuEWBEMmLZ5A+apR/uYSqzMIxGah+MT6W3RcjmJUAo01N7WLSmXksAn9UkUY2IxbOw/fGR7N9XxcLH1kcSdjr77O8aYmlSOiaVE7nkEI50Og4jDBi31RpZgAaJ7qXdzTx6PCdjENiNPRUEJHEzMPo2Lt3s1yWeLsjDr7lx7A3ICS958ueuNIeaWo1HwA5sk+hJrR133PaQYqKvUjzP4j0dXubIcfJvLBZsy679yNf4A6RO+Mh06zQK9ywAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
Adding and subtracting the square of coefficient of x we get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYoAAABHCAYAAAAZfwDOAAARrklEQVR4Xu2dCdR21RTH/yFkyrDCMmUmY6akMmWmQkgoLJIv87BIhrIqlUrmEkIiIk0UMiaRoUwlS5YhqWSRWSXD+rHvWrfne573vfecfe499333Wetb3+rrnn32+Z/znH2Gvf97HUUJBAKBQCAQCASWQGCdQCcQCAQCgUAgEFgKgTAUMT8CgWkhcANJG0m6VNLZki6blvqT0TZwbg1VGIrJzNtQdJUjcBVJayStJ+kMSZtLepakgyQdKuk/qxwfr+4HznOQDEPhNb1CTiBQFgGMwi8kfa3VzJMkHS3pmZKOKNv8qpEeOIehWDWTPTq6shC4qqTjJZ0u6c2S/mndu5adLv4uadPWv6+s3g/Xm8B5AdZxohhuEkZLgUAqAteQdKakDSXdQdKFLUFflnRnSXeT9IfUBqLe/xAInMNQxE8hEJiLAJulDSRdPAA+60u6XNI/Etq6uyTqf71V95p2orhC0v1MdoLoQaoEzoPAXKaROFGUwTWkTgMBrm7eLYmFlofifxVWeztJu0ni73Md2tpE0mkm80AHeaVEBM6lkB1IbhiKgYCOZqpDgN35UbZgv2LA+/0nSjpAEg/RP8hAZV1Jx0r6q6QdB9S/r8qBc1/EKvw+DEWFgxIqFUeAxesY8yLaZYRFdntJ+0jaRtJZib19uaQtzOPpb4kySlcLnEsjPJD8MBQDAR3NVIMAO3FcSQmoYnePx9AYhYX+JZK2NIPVR4enSHqgpFdVHHAXOPcZ0cq/DUNR+QCFeq4IMN/3lvR0SQ+QdJGr9H7CWEg/LOk2kh4mqeupAAPBH9xkmzeV+0j6UUWP2YFzv7lQ/ddhKKofolDQEYHHSfqYpCdLOtlRbqqom0r6psVIcMJYLroaz6cHSTpE0r9bjb609SifqotnvcDZE80KZIWhqGAQQoVBEGBRJmANA/H8DovyIEpJ2lbSkfZesZTx4uRxmEVmNwaF3+91JOEm+6KhFF6mncC5koHwVCMMhSeaIatWBJjn8CFtLYlrmgsqUpRoYB7WCaTjOuxPc3TjmuoLkh68QO/XS3pTBX0KnCsYhBIqhKEogWrIrA0BgtHgSNpD0v5OyhENTezCDpL+mClzYwuk28/eUDLFjVY9cB4N+rINh6Eoi29IHx8B2EBPMIqLe2fSXEC+d0dJN5b0CLvygTrj95nd5Hd4uKTHSLqXpPMz5Y1RPXAeA/WB2gxDMRDQ0cxoCHCdw2liT0l7ZWrBzp/Tw3mSPilpMzNAuYYCtdiNn2q04a/N1HOM6oHzGKgP1GYYioGAjmZGQwAvp8dKYpGHpturfMrZULAj/6IR/N1T0u+8FB1ITuA8ENBjNBOGYgzUo82hEMBT6Pvm6USQmmfxNhToxnsHwYA7mYeTp74lZQXOJdGtQPZKNxRTSmeImyOBYNczRtCvSGJ88Em/tbGbsjiVJq6rYFq6qfBqC0yDVwnPIs9SwlDcRNJPjAOKILypjHXg7DmzKpS1Ug3F1NIZYgieIOlDkv4i6f1mLKCjhlaae/FPSGJxekeF86hGla5md/63lXQXhwfn2T6WMBS08RlJD7dH7XNqBHZGp8B5AoOUq+JKNRRTSmdIsBTspTCKNpnL0P9dkp5txoEAsfdIeoFF5eaO+2qov5Gk79m9/1YFOlzKUOxsMR/wQL2zgN7eIgNnb0QrlLcSDcXU0hlytfRrST9szQ+O8lA63NXcObmOwiuG66g2dUOFU6oalWCFPVjSi83oeitWylAw5ryrkLnu0RVFkC/CL3D2nlkVyluJhmJq6QxvYakt2/fRn5a0nvnqL8f/U+G0qkIlYieIxL6/pG8X0KiUobi2pT1lc9BsFAqo7yYycHaDsl5BK9FQgPaU00Ze1x4zPzDxKN0xZ32z2OIgwFwokUu6lKEAt+ON+4k4DUgDay2Bc60j46zXSjUU82CaStpIrpjYAcPrQ6BYrQWHAby00BMiuBtK+pmkt9rVyZh6w5v0Y0nfseQ+Ja7rShoKuJsIDnyhXZ+NieVSbQfO444Mm6BnSPqzqXF9c4T5qbdaq8VQeKWN5OGZhZCEMaSgLFFeJuk1du3QjvjFE+ZLTnfWuf3ASBA9TCpPvHS4Hru6pH3twZ0+QMKXUnJ1o02oME4yVlZ+SCVKSUOBO+/Rlq8Cx4ZaS+A83shsbq7fJN9qgjPZsJEeF+cY15PoajEUXmkjb2SLNT7uHrQNLLjkEsANFq8mxuPzkkhGT96BZifMbv2pjh5Puf2g/xtK4nqsXTAWpxitxaaSzk74HeXqRpOM90GSSrKqljQUzanytJl5kABn0SqBc1F4Fwrnyo/T8uvMMLQ/ZJ1g3vP765oMa9lelDQU95D0PkkPHTHdJAB4po30WMTagwIB3Jm2++Xhlf+GLppd9UPsQ4wJP0gW5UuWHdFuH+T2gzgPYj92swnbbvWVxqqKuy/eW31Lrm60B8EeBH5cjUEtUaLgcMAbAjEav3Vu4GaSzpX0G6MeGStd63LdCpyXQ6jM/+cUQXZEGIyZI+3C75KgTZJzcdp3KSUNBYE4/Ei5b2fRGKN4p430WMTaOBA5zs4UI3CZJOR/1K63vmXkczCeHiXp544A5vaDRZLYhPdaEqC2agQOcvwlEporlL4lVzfa+5ykR5nXGPxJXoXYBu7l0ZFxYY4zLryHcP22uwVM5raHxxOpTdeXdKcChihXv6Z+4OyFZD85H7FNCpvx2Stw5gxzh5uJ5/UTu/jrkoaCVqEkYDLhy06EcU7pS3FRIm2kxyI2iwEnBqirOSYST0FhXG5pCxFEdt4usrn9IPkPfEREic9GD3OnToQ5O56U+/Vc3YijYa5x9IZcrx2fkjP/hqyLi/cZ9k7F3OB0UVsJnMcZkQZ3vCP5HbLBbBfc6pnznHLb19dZ2nY1FDeXxINwSuEHyyPn9pLYJaeUvhQXpdJG5i5iKX0vUadkP6Df5thLik9OFn1Lrm5c23GdR8Qwu388saZWOKnwGHlf+4PRqK0EzmuPCJs+4qL4O7XwLkk+kkWeeo0hgNYHavcrFhgKNpdslHj/zC5dDAU7eXINE/yTU0jxiDcM95p9dsh9KS64Py+VNjJ3EcvBz7NuqX6w++WRDbdeDEVDSdJH91zdcARgR3U7SbdqndL66FDDt0Rm875Xq5t04Lz2LCGJFVfJ7PpTCwaCK1uuj+YVXGD5fxctMBTNJoPvuJoazFCkdni2Hrv85i6e1/quu6SaKC5yF7E+BrIr7l2M/ays3H7M040fx5F2jQb1ROoDb65u1D9LEu8/GIvZx74uuHqPU8oY1W4oAucuM8n/mwZ3Th3zThSTNxRA1gRpEScAS+pxdsLAOi4qQ1Jc4IeMu+qioyOnG9zPeFy+dIHCLDKQuaUsUF7Taox+4Jm1oyUJWmo8S+vW/JDA0iNNqdeY9JXTGAqMLg+TqYVr2zX27pUi47uWzW/eZgODHDinoJpep3F0uHiZqydamOSJojEUj7ck9xyJoCrg4XOphWUW0pIUF+RC5uGdeIB5pYuhuNziHS5InwvZNYfuxzb2uA3b7XJ0GaV1awwFu3iuSz3iXbIHJEGA14mC6zdYh1NONajNyR+K+0WGInBOGNyMKs3bECLwvJvdsDZvGPw7b1yzj91JTadOnpTG8OLBnZLd+hsyiNrGpLjIvRZJwa1EHc9+bGGnrF1n4mUIElzOaMzrW65uBCPxRkEeinijKDF7/i8zcC6H7VKSWbPZRLCecmKYjbFhI80bBjQej/RSsYuh4KiDiys7wZyCpXujcZHk3AEPQXGxqJ+5i1gOfp51vfqBC/LT7ITYfrjmRAZPEXQnfUuubswzqEXweCJ6/Ly+ClTwPffM3zBqeTjKcBCorQTOa48IJ1hcs3lITi3wNuHWyhxeVPazGAk8+7iCaheudnGnhplgj1QlZut1MRTUyXGPZTHhzh7OHSgJ+pSxKC4W6Zi7iPXpe8lvPfrB3TcRoiRYmvVuur1FluOB1rfk6tZ2LWXHtch7pK9eQ36PoeVtgN8OiwGRtrWVwHntEWG9Yq3M8Xoi3QDvm0sRWeL2SqgBp3nmSbvwb3h9Qq/vFkPU1VCkTlIC7lAakqqUCNmxKC7CUCw94htIOlESf8/L68yOCuoUEi31LbmGgvYgT9zSdman9lUg4XtwaIjZEqqvVaW5PgALTkZ93vA82u8qI3DuipTvd6zbeJASlQ1bQHNDw79zvY+hem7PMIQlNSxpKFD2CCOGg78opYxFcTGGoRiStjt3MYZCYClWVoKAiKkgqrxvydWN9qBBgecJYwULa8lCdOyBRhcyG/yU2m5zfYDxmUfTkCrXu17g7I1od3k8CUAkSrpfTvWs5RgNuMdwXnBlty5pKJjgWD3I7XKUHoPiYmhDUZK2e15fPBbj7lO635ceuhGns7eRKb6tX/O9vibo7GRJUG7M82nvJaz1MVdOXBvwTgFfWYl8Gqm6tesFzh4opstg/cbzidMzc4QTPKwE7qWkoXBXtgKBHovYvG6UpO1ejYaiyefARoUjeKmCbB4MCS70NBTETnzWSDU5GdVaAudaR8ZZrzAU/QDFhxmPA5L2eFI/l6TtntfDUv3oh+b8rz10a3bkX7XdVo6X3aI+cWLGBZfjPm8KnoaCN723SIKyHe+VWkvgXOvIOOsVhsIZ0ERxJWm7E1WadLUmehUDwYLepIr06hTGDAPBYo6zhreh+LjFpjRZDb309pYTOHsjWqm8MBR1DExJ2u46ejisFsxrYn8IOCrhIgtVCd5Uv7TgJ09DgWssOVxwP8Y19sJhoevVWuDcC67pfhyGov6xy6Xtrr+HZTQk+94+lhOD3BheBW8uDHuTOY8oWU9DAXkmHEp4s8AcO8/92KsvHnICZw8UK5cRhqLuAfKg7a67h+W0a6heuMYhetyjsNuHNBJPqibI0NtQ4NILtxKP5Ht6KF1YRuBcGOAaxIehqGEU5uvgRdtdbw/LagbFBFGrvCcQuOnxToHBgSQPHp2meBuKD0rawTL0daXiL4vk0tID5zHRH6jtMBQDAZ3QTFfa7gTRq6YKj814EBHLc0pmr8ltgWcTwYbt4mkoiGiH4wfGW9pyYf7M7HeX6oFzF5Qm/E0YijoHrw9td509qEOrjY1Q7zDLyZCqFbxGuKpCcgiNfClDsZUkPOBqd4udxTFwTp1ZE6kXhqK+gfKm7a6vh8NpxAKP+yokhXg/XZLY9GYW6f2rOfXxrOLt4iSLjt1f0jmJ7ZBGk/SnXJXNaytRbPFqgXNxiMdtIAzFuPjPtl6CtruuHg6vDQy3x0h6jiTu/z1Lw6CKzNyAO4wZbxJ4ue3kqeRAsgLngYAeo5kwFGOgPr/NUrTd9fRwHE14zCZHAHN98yVS2KZo52koDrDrsU2NSDNFnzHrBM5jol+47TAUhQHuKL4kbXdHFVb0Z1tLOkHSdgvyP6d2Ho8fTgG4yuImOvt+0VUumfiQA9PtLl0rVfhd4FzhoHioFIbCA8V8GSVpu/O1m74Edv7HWcY7rohy2IxBg4V9d8tkxkMuBVdc5BL70Der3tst1oMcx33r1jQ6gXNNo+GoSxgKRzBDVNUIQIdBhsV9JXHNU0shyhvXXTydDq1FqQw9AucM8GqtGoai1pEJvUogsMZoPXirSPVM8tSryWeBN9a2c1LKerY1pKzAeUi0B2grDMUAIEcT1SCwriR4n8hrTM4HT6r4lE5yfbWzRWGfnyKg0jqBc6UDk6pWGIpU5KLeVBEg+pmsdKcbb1OJXBVdsCG4jkBAkv/glbXSSuC8gkY0DMUKGszoSmcEcEXGWBxi0dadKzp9uIkkgut2lXSkk8waxQTONY5Kgk5hKBJAiyorAgGy051oGQsPH7BHBFUeK2kvSUO2O2AXr9RU4DwW8o7thqFwBDNETQ4BdrxQerNgHyyp9DUUsRa4QnOSwF13tZTAeeIjHYZi4gMY6gcCgUAgUBqBMBSlEQ75gUAgEAhMHIEwFBMfwFA/EAgEAoHSCIShKI1wyA8EAoFAYOIIhKGY+ACG+oFAIBAIlEYgDEVphEN+IBAIBAITR+C/eVYGkwJ5ShwAAAAASUVORK5CYII=)
Use the formula (a+b)2 = a2 + b2 +2ab to get,
![](data:image/png;base64,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)
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)
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)
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)
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A3O7uhlgL94HqMP1p39bUbFONOrluythlKYEpAn+H01SFXCcRJoAI9Tk71qSqBoiVQgV709NXOVwnESaACPU5O9akqgaIlUIFe9PTVzlcJxEmgAj1OTvWpKoGiJfA/4+PpZtzozBsAAAAASUVORK5CYII=)
Square of a number cannot be negative, Hence roots of the given equation do not exist.
Question 2.Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.
Answer:For a quadratic equation: ax2 + bx + c = 0,
The roots of equation is given by formula
![](data:image/png;base64,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)
i) 2x2 – 7x + 3 = 0
According to the quadratic formula; ![](data:image/png;base64,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)
![](data:image/png;base64,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)
∴ The roots are = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
∴ The roots are 3 and ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAiBAMAAABhHd5lAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAAAA6AGa2OgA6OpDbZgAAZgA6Zrb/kNv/tmYA25A629vb/7Zm/9uQ//+2Yg8ohgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAASUlEQVQYV2NgwAa4DBgYuAWBBA83kGBAIyCyggFYtYIFBQUFBXDLgmX4E4UZGL5PuDQByH4CIrYCMS/Its1gBm8DwzdBwQaEKQBKFwtgUKi3iQAAAABJRU5ErkJggg==)
ii) 2x2 + x – 4 = 0
According to the quadratic formula; ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
∴ The roots are = ![](data:image/png;base64,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)
=
= ![](data:image/png;base64,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)
∴ The roots are = ![](data:image/png;base64,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)
iii) 4x2 + 4√3x + 3 = 0
According to the quadratic formula; ![](data:image/png;base64,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)
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAK0AAAAbCAMAAAA54v8ZAAAAAXNSR0IArs4c6QAAAGZQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6OjqQOpCQOpDbZgAAZgA6ZgBmZjqQZmZmZrb/kDoAkDo6kDpmkLaQkNv/tmYAtmY6tv//25A625Bm27Zm2////7Zm/9uQ//+2///b7WxJ8wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB7UlEQVRYR+1W2XLCMAyMgZJS0gO3xcUFH///k5XlxPGVuM5LmU70wDCMV1qt1jJNs8aqwKrAP1KAkzHuvi39fr57jiNB9Xr7M7aMbC91xcW+7vzy03yTTJGVi4codoL6Aqyb5ioQqwTJNqmg6bHUfIjSHzALBnk0raRbCVLdW1JAPZfuTISSB7AtMy1yYlT+fQQg0HnzaVJNBjuKhK3YXgrWjVDczYLFjpctKd6CHmQ6FWTOhGJ3S9nyB3pS3QwsRn31s9B0FyljGPD4x1C6ASRb6Fm2w2yYW+BOTTPzlK0101AjhTmUpiiFesEVojpCImLIwC2MDAEPhArLx2kT4nUCtjwwm+rg11Qmp8iI+j5jiXF/yTZ0QmKMnCMtSFPTOJ/ZnWZ9YARssb8ZJ/goFM9rVhB/nVgGxUBQr5ED5AZh1mR8y7A/4TrIw3qUptuL/+wiexfIYIwJAmDWgS0Pms21mbBlxnz4MRMDCrrC/QWTNMzGJq2TC4vbA5k5iKfuaLNNhnuVeqsauZrMAxcmGA4A6IqUEObcbr+gE2AfTtb2QWCwfcPiexpBzUq297AvJQ/X8oocUfBf0brerAS3LYdcsG1nrRuAigb3D6iu7h2yWNXltFuWq4KupkvIwkbIiLc0VwXd9eiqwB0o8AN2wiMZ3iL7+QAAAABJRU5ErkJggg==)
∴ The roots are = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
∴ The roots are ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJEAAAAmCAMAAADzyYncAAAAAXNSR0IArs4c6QAAAG9QTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6Ojo6OjqQOma2OpCQOpDbZgAAZgA6Zma2ZpDbZrb/kDoAkDo6kGaQkJA6kLaQkNvbkNv/tmYAtmZmtv//25A625Bm2/+22////7Zm/7aQ/9uQ//+2///b44UixwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACR0lEQVRYR+1X20LCMAxdgeGNoYIg0yl12/9/o016WQZLbaP4RF82ynp6epqTtEVxbVcF/kGBRrl2mblY+H57YGZ8e/d/6FJMKh++veEIdY8DjWYe2OVxy4fvqmduCr0qClDWPMzL4phHxX4tgK/5/fg4FCC5noGI/QaJ5bZ8eF7Uon+xqrhPtGTfBPDMGhojSHuPhPqN3VaRSPnw7XIyiupbEzUa/6qVcp8IglsAz80Cwnjve1jtqaXHkgC+5hzULD7B+936UHhGXZUQ212liFXy4buKc1q33j+h6b37IZBSEgDlLYBvl+yyG/T8qNUpbqOhI4CPhEZ7d5YRJ0iehxRNEgL4vGAdMdJKzfaGda0WR/OOfjTP+Y5srQA+adVBh4aYDd61Ko0n9XxnHsAD0EzfoJsAPm8IWTJGiA2ZBuoeZELS50gJ4MMQkwddw4AOv0bnI8IIg9yWCHjFfI59tgieMkqHF2y0qyiwN5j/kAyo1W9Cn2ckgBcMwdkw6VgKuHVgMNL3C0ZM3WHKxJCPcHYMILtNkJtJnx8vgE8qDIHfUBNAHv1QrUzSwq2ry377OvT5EQJ4K3xiox+bzFNiKnLRo9D4ri/gSeDZUjjBMlITuDUJ4EdmPcE1ZZymu7Gx03QVwEd2Gg4iI8ScBTvCEvj4FYMekyO3Cl4wAXzcoHDcDgk45XR0yk0CH1sFvcmKJIpf8zh4/kbVkNQguomAYgJ4NmnAEux9JAr8g+tE8F+TF38wf7gZBWZprh99dWF4AaPrkD9V4BurMjhjLuuNFQAAAABJRU5ErkJggg==)
iv) 2x2 + x + 4 = 0
According to the quadratic formula; ![](data:image/png;base64,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)
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJcAAAAZCAMAAAAL+0zjAAAAAXNSR0IArs4c6QAAAFRQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6OjqQOpCQOpDbZgAAZgA6ZmZmZrb/kDoAkLaQkNv/tmYAtv//25A627Zm2////7Zm/9uQ//+2///bQwY89wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABhElEQVRIS+1V23aDIBCUJI1NpZWW0AD+/392WSAWWEF96ek57kMe4l5mZ8ex6444GDgY+P8MSHa+//EWmjF2+spByGsLlnktilol8FyVo+gqCYmW5+xMYqgPsZxYpo3L9KtxOQCKjWlP+14nQ5/vqzf/1dnyj4gLznT6vD3qq+gcF8xtSYzCZXpWFaYcdMDlmNCsJRaZ31G9iNHyahmByw1Tl2UO9OURcJkezmT6eCUJGveRnHlOiLQ61U0ijiDLSlw4TMdlyiInjoALiWi9OaXILYcJMy5SBCWugvW0DscALjVCa4de1b3IJyWBm2y9I9EnaeoMCWOE1rj3cy55EMKqcJP5ZVh3Rxw2B60Zf0dMVazqRShUnaZI9x/+LEdxxwwXXelxOar0Gx/MslF48UmPK0hqEsBXy5/m588quAsYTNWSQhXc9NpJVtk80J3gMrfvug/BAqgUL5D4gjj3ajiSc9Mdny/LM99vf1lQLXuqVrX2SZPYM2Bf1QZYR+rBwCYGfgC6qhSZ/zm8OAAAAABJRU5ErkJggg==)
∴ The roots are = ![](data:image/png;base64,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)
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARMAAAArCAMAAABsKZz+AAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjqQOma2OpDbZgAAZgA6ZgBmZmZmZma2ZpDbZra2ZrbbZrb/kDoAkDo6kDpmkGaQkJBmkLaQkNv/tmYAtmY6tmZmtv//25A625Bm29vb2/+22////7Zm/9uQ//+2///bJN/gigAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADmElEQVRoQ+1Z65abIBCO22zvTXrbZLttbGPbGH3/ByxzQ0RRlPGcbY782c0MfMDHXGDcbNa2MrAysDKgxUCRZR+0sAI4eZYdFp5CF764j8Grj3dZQjveIifV5xjmQn3qm+SkTPKv2+Tk92nATq6vRJuzI1oJCW6Gk9yJAfW3c5iTYnuxyhL/dyQo+N85kaPeuJxQOLEqpuD6Emi7vjO7rnZZhiZRGCdDifmDahAkcWIS43CbG+lCwGbNrbxj+slvlxMIJ46KVlE/vpFNV/vTprwDD7q+PiMNVg2CJE5+DZjoXDpwXBi4zcmfhiOXEwgnjoqWUjxAOrH7pRhSP55YQmoQTOaka6JJm28GRwL79xP8Da4AjSMFh5N21/IeN285Yes4HkjCatRH2kmv22LGa3stGD1IrcOOMRYG5pGCyDPBRoUCdIUe3+HbSatr9fFcHx+eLrLfksaZnygRdTwnAbc1Jup5LVgeOGrjsMOchIFlnCDKTL12Ap3ZznJzIiWloFZXilBb4UR04juijvedfretny5drzVhi9N/cw0I8xIGdscgEnf1OfHsLP/04rzh24mfd8hE0GfASpC5Jsayx0yIsb7bYg573zkNskaOaVF3yTCwZUUQsWsn78ibkO0EnMGcFeYd77lYZIYvZIGcTxgCCQ4A9UguHnDb/K0JZ2SinoPbRyU7bJ+NRAA7XSxiDyc96MX27/Bjx2WVriaOBAWRMdYPZTCMEmbnocp3n6j3azdGNsDNfgmRZup9F7t5p9p/H3nV0s0EYxAnKishQSQnnfRgj6PltRBbcQfWYcOxhJfl5w3vnC2ik3eGMQu75bG5Q/ooThzflGtEtf/xpcdrOXM2Djuy/sbpPWA7TnIxriFgJ61J+Ko+l5B4O+nOkH4cgVUPA8fVlBIImchJ67qYfhzNwicAPzdO0tjXGb3WY3V4nI1irgHckuN2ew2qwGPFCS19iMZ0/ADyfODZJ74O1GJA1cTdRS0GrLXzFWdlYGVgZWCQgU7xM7oamkQsP+GTMDqDlUBt8ROflqbaGlsNTdsNfy9JA/FHa4JiybLaHbA6DC2mGpq2Hf4gkgbij9YEpa8V1e6roQUbCRZs8kFEdQpNUCl+gqVgG6iG6mzCfhDRgSMUTVApKlg7Wb7KYD+IaHKiCCrFT4glaCnR1dCkDUWVLafOoARqi58/IbxC3rGfL6auaFJ/+iCi3BYBVV7jCvfcGfgHfxJahmv+O40AAAAASUVORK5CYII=)
This is not possible, Hence the roots do not exists.
Question 3.Find the roots of the following equations:
(i) ![](data:image/png;base64,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)
(ii) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAP4AAAA0CAYAAACq5YWDAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAARQSURBVHja7Zw9TtxAFIDnABwgSEw6ujRg5wrERFwAVnSIClmKOICh2yIcIOnoQkFNwwm4QQpusHdI4l3seJe1d37X4/FXfFKkwGb9Zr55M2/8Im5vbwUAjAuCAID4AID4AID4AID4AID4AID4ADBu8afTy51sTzyL5Ox+rIGOLQa6z8McGJn45YBPMlkIIf6MddBji4Hu8zAHRiR+vcKXg10xskGPLQa6z8McQHzER3zEH9sZvyjOdxMhXsc86LHFQPd5mAOIj/iIj3yIj/iID4iP+IgPiI/4iA+Ij/iID4iP+IgPQYm/9g63QTK5OYk9uLHFQPd5mANqnCXi3iQe1e910faZTHrER/weuTqS16bxCFJ8AFCTPk3THy4Xwv8LbvJ6XhS7iA/x1k72sufL6XRn9e8OpXgRQs6yvDgw+exUpr/a5LHhZpKclLLLo6vr6s+uxG9+9tbP+ADbYt1Et5V+sZWWdz6OI0WeHUghZlWh06X4Ktke8SGyWspisruQfi6nzB5WdxE+diguxVfJ9og/IlQKQZsmS9Bb/iqLptmjrvTNAlsrDq4h6wVq5VjiSvx6UdmQ7REfIl3cktfTvNi3WkQ8ZPtKcNNKvOoCpvL7ZLkBZzloynq6v8h2/8Z1TaHP99leaa6lnx59id9V5FQSX+WLhYaPiRRbDGwXwZDHsJZeHr4cH8vvuvJsa6u/aSdgs9XXyfZs9SFo6q1xRxZrSl8V9nQyn4ts71N8tRi81Tc0npkJBsFSZbG2nUg94d+kX5VF9xjn62xvI/6mGDR3czoLFhMMgmTTfXSb9Cq/2/rvSflgev3nQ3yV5zDJ9ogP4Rbr6ObzGoNBBcK0i2mo2DRh+CKfZF9TKZ+WioIyfWq7PiuzVnlunmel+c/LmUzO7jZlp0UWNH8BJ54ah58YDO68h/j9iV9vK9d+l/fb0c7uvA1b0/l4W17LRVHj8BSDQUnvuosp1vOvT/HLppXTPD9Y+i6peFxXgGpWpKsdQVmFX7xZxzj2idoZw0Pnk27hw0cXk/ZZq8c4tMUknC3p++/TdjSrFwT+N54wxW8bVJPJXn6O0fWKpy4mF5O7D+n7yvbtY1Rl8OXv0/U9be/aYQviu+p80hXfdxeTrXB9SB9Stm/WHz4mX36uCoz4Axd/KfMadD6ZiO+7i8nmjGsbBydHjgCy/XLhUc6So/wC8SMTf3mgzTqfdMT33cXkZsLbdYCZoPs+9rZ2QuvO8ogfRcbX73xSuYpqK+6EKr5pB5iLLsGQZVmtxSB+BOLbdj6ZnvG7FoQ+Mp6rOMSU7bvEby54VPUHJr7Lzqchi++6A8xYrJ4zZH6UXJRv3eX5zYdmbKp7edV7/OqlHu7xAxTfdefTUMV3HQebukLfonT2ra9ZlGze3IMexHfd+TRU8X3EYajZvnrmSfb5m5Tid7Oi3/Xuvem7+hBAcQ8AEB8AEB8AEB8AEB8AEB8AAuAvk29/PD+PlT0AAAAASUVORK5CYII=)
Answer:(i) ![](data:image/png;base64,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)
Taking L.C.M and then cross multiplying we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
According to the quadratic formula, the roots of the equation: a x2 + bx + c = 0 is given by the expression
![](data:image/png;base64,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)
For the given equation,
![](data:image/png;base64,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)
So, the roots are,
![](data:image/png;base64,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)
∴ The roots are = ![](data:image/png;base64,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)
(ii)![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJ4AAAAZCAMAAAD37mcpAAAAAXNSR0IArs4c6QAAAFRQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6Oma2OpDbZgAAZgA6ZjqQZrb/kDoAkDpmkLaQkNv/tmYAtmY6tv//25A627Zm2////7Zm/9uQ//+2///bQqrffQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABiklEQVRYR+1VzXqEIAyE3da2tnWrlRbx/d+zTAJCP1Fy9EAO7mGTzGTyg1LNmgIXUMD1Wj9fgEeZgnsflb2Nl+XniS0vl6Zn3g7Fs1prmbbJc+l8kHheUtw6aH2fd1TscarJU1sHCT94uh7Zjf5Uykr5JYR1ePpdhx0/c1LoBF0Jr2abJ1C881TQoZQjIdAGWMLyOnpgMFfQzp7iE1IWccwUySM9Iik0QqCP6yHW92juswEp3BV9Jk+ASxEnkIRgUTmrIDNG4K/ruSx7+3oVRIN90KEesXTECTGptRPKJyuPcETISSJHYTiKqZaOPfOIouM60AVYOv9jIOGB7YIJ4T+9kEsgIHcL6h8j8v+0YxFGuBpEAAixubyoNvasTpDkqEdMnJiHO21urbmb4GE1CMt9/AjKY8F4zGsRBiNqowphoaq1ZwjhXtJhecx+jn2yc6MrGbpVieCHEVeMxi4MhIDehoClZSxDj4HgsNPZQbOqEaGHrLb0IaRZ2F7A7EpUy2oOTYGmwNUU+AOJmxlBA1my9gAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
Factorizing the quadratic equation, such that the product of two numbers is 2 and their sum is 3 ( 2 x 1 and 2 + 1)
x(x - 2) - 1(x - 2) = 0
(x - 2)(x - 1) = 0
i.e x - 2 = 0 or x - 1 = 0
therefore,
x = 1 and x = 2 are the roots of the given equation.
Question 4.The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago and 5 years from now is
Find his present age.
Answer:Let the present age of Rehman be x years.
Three years ago, his age was (x − 3) years.
Five years hence, his age will be (x + 5) years.
It is given that the sum of the reciprocals of Rehman’s ages 3 years ago and 5 years from now is
.
∴ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGkAAAAiCAMAAABrwwenAAAAAXNSR0IArs4c6QAAAGZQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6OgBmOma2OpDbZgAAZgA6ZgBmZma2Zrb/kDoAkDo6kGaQkNv/tmYAtrZmttuQtv+2tv//25A625CQ29vb2//b2////7Zm/9uQ//+2///bQkcfMAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABL0lEQVRIS+2W2RKCMAxFg+KKCwouuED7/z9pC2UtbZpBH2TokzOe3JukSQeA6fxbB+IFJWMK3WYTzyM4Uegu+4SE4EShdZbiBKS8uspjdKJMEwCF1mbP87aucy7myZl2ZXl4dLUfyE1OQxpI6F4sxyQ/s0i3rP7s/MhJLdRKG+pRMXi1hJpwMStBcLJ3D82D4IRq2YExOg1siS1cPojOLy1Ium97Gg4mQX6KIEVipYy6vwR9fyvBVHyopJ0islXPjudZ1rSzkwyTguyQra+btJWZedwadOkkuod+6SiWh/4bgAXFqyEsxQKaW6LRIuEl0sFSkAXa/VtiK7ounIfoAEnBx21/8V9nUZU6bBeZs2zQhZO8bqSmUjCe31nQGjXblDdo5RTiO0Fbmx/u79elP7xnE1jwDSx+AAAAAElFTkSuQmCC)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 3(2 x + 2) = (x - 3)(x + 5)
⇒ 6 x + 6 = x2 + 2 x – 15
⇒ x2 – 4x - 21 = 0
⇒ x2 – 7 x + 3 x – 21 = 0
⇒ x (x - 7) +3 (x - 7) = 0
⇒ x = 7, -3
However, age cannot be negative.
Therefore, Rehman’s present age is 7 years.
⇒
Question 5.In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.
Answer:Given : In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of her marks would have been 210.
To find: her marks in two subjects.
Solution:
Let the marks obtained in Mathematics by Shefali be ‘a’.
Given, sum of the marks obtained by Shefali in Mathematics and English is 30.
Marks obtained in english = 30 – a
Also, she got 2 marks more in Mathematics and 3 marks less in English, the product of her marks would have been 210.
⇒(a + 2)(30 – a – 3) = 210
⇒ (a + 2)(27 – a ) = 210
⇒ 27a -a2 + 54 - 2a = 210
⇒-a2 + 25a + 54 = 210
⇒ -a2 + 25a + 54 - 210 = 0
⇒ -a2 + 25a - 156 = 0
⇒a2 – 25a + 156 = 0
In factorization, we write the middle term of the quadratic equation either as a sum of two numbers or difference of two numbers such that the equation can be factorized.
⇒a2 – 13a – 12a + 156 = 0
⇒a(a – 13) – 12(a – 13) = 0
⇒(a – 12)(a – 13) = 0
⇒a = 12 or 13
If marks in mathematics is 12
marks in english is 30 - a = 30 - 12 = 18
If marks in mathematics is 13
marks in english is 30 - a = 30 - 13 = 17
Hence
Marks in Mathematics = 12, Marks in English = 18
Or
Marks in Mathematics = 13, Marks in English = 17
Question 6.The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.
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To find: length and breadth of the field
Let the shorter side of the rectangle be x m.
Then, larger side of the rectangle = (x + 30) m
we know,
By pythagoras theorem,
(Hypotenuse)2 = (Base)2 + (Perpendicular)2
Diagonal of a rectangle is √[(length)2+ (breadth)2]
Diagonal of the rectangle = ![](data:image/png;base64,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)
It is given that the diagonal of the rectangle is 60 m more than the shorter side.
∴
= x+60
Squaring both sides, we get,
⇒ x2 + (x+30)2 = (x+60)2
⇒ x2 +x2 +900+60x = x2+3600+120x
⇒ x2 – 60x – 2700 = 0
Now for solving this quadratic equation, we need to factorize 60 in such a way that the product is 2700 and the difference is 60
⇒ x(x – 90 )+30(x – 90 ) = 0
⇒ (x - 90)(x+30) = 0
⇒ x = 90, -30
However, side cannot be negative. Therefore, the length of the shorter side will be 90 m.
Hence, length of the larger side will be (90 + 30) m = 120 m
Question 7.The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.
Answer:Assumption: Let the larger and smaller number be x and y respectively.
Given:
According to the given question difference of squares of two numbers is 180. and the square of smaller number is 8 times square of the larger number.
So,
⇒ x2 – y2 = 180 eq(i)
⇒ y2 = 8x eq(ii)
putting value of eq(ii) in eq(i)
⇒ x2 – 8x = 180
⇒ x2 – 8x – 180 = 0
For factorizing this quadratic equation, the product of numbers should be 180 and their difference should be 8
⇒ x2 – 18x +10x – 180 = 0
⇒ x(x – 18 ) +10(x – 18 ) = 0
⇒ (x – 18 )(x+10) = 0
⇒ x = 18, -10
The larger number cannot be negative as it makes the square of smaller number negative, which is not possible.
Therefore the larger number will be 18 only.
x = 18
∴ y2 = 8x
= 8×18
= 144
= ![](data:image/png;base64,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)
∴ smaller number = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACEAAAAZCAMAAACIE7edAAAAAXNSR0IArs4c6QAAAD9QTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6OpDbZgAAZrb/kDoAkNv/tmYAtv//25A62////7Zm/9uQ//+2///bnKQzbwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAiElEQVQ4T+VSWw6AIAwD8YU4UeD+Z3UwlElI/DbugyxQ2rUgxP8qmOU27aY19YeUsqMWqyC8ztuAp16rrUYcarN0EWZcrLy4uUpGZKU3BCQVwImoiJ5xuPE20FYJJs5Se2EcwQzl3ZscwAAsj8Jh+x1ju3QaHBRtSuVRwSRXyJ8tRkTt9ru/8gS0ugVmNeLm8AAAAABJRU5ErkJggg==)
Therefore, the numbers are 18 and 12 or 18 and −12.
Note: As we need to solve this problem using one variable, so the variable y is written in terms of variable x.
Question 8.A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.
Answer:To find: Speed of the train
Let the speed of the train be x km/hr.
Time taken to cover 360 km =
hr, As
![](data:image/png;base64,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)
Now, given that if the speed would be 5 km/hr more, the same distance would be covered in 1 hour less, i.e.
if speed = x + 5, and
![](data:image/png;base64,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)
then, using distance = speed x time, we have
![](data:image/png;base64,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)
Now we can form the quadratic equation from this equation
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, cross multiplying we get
⇒ 360 x - x2 + 1800 - 5 x = 360 x
⇒ x2 + 5 x – 1800 = 0
Now we have to factorize in such a way that the product of the two numbers is 1800 and the difference is 5
⇒ x2 + 45 x – 40 x – 1800 = 0
⇒ x(x + 45) – 40(x+45) = 0
⇒ (x+45)(x – 40 ) = 0
⇒ x = - 45, 40
since, the speed of train can’t be negative,
so, speed will be 40 km/hour.
Question 9.Two water taps together can fill a tank in
hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
Answer:Let the time taken by the smaller pipe to fill the tank be x hr.
Time taken by the larger pipe = (x − 10) hr
Part of the tank filled by a smaller pipe in 1 hour = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAAiCAMAAABV5mjpAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6AABmADqQAGa2OgBmOma2OpDbZgAAZgBmZma2Zrb/kDoAkGaQkNv/tmYAtrZmttuQtv+2tv//25A625CQ29vb2//b2////7Zm/9uQ//+2///brA/0uQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAW0lEQVQoU2NgIA4IsILViTIyQhjiDKIQBlAMNwOhi5GRgziL4KoYoYBEbXiUSwHdLgV0hwy3NJsQuxQPUKksL4skWIcMJ9h9YsJcgiwSfJIMAswiMpxM/FhMAwA36wMRTHi2kAAAAABJRU5ErkJggg==)
Part of the tank filled by the larger pipe in 1 hour = ![](data:image/png;base64,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)
It is given that the tank can be filled in
hours by both the pipes together.
So 75/8 hours, multiplied by the sum of parts filled with both pipes in one hour equal to complete work i.e 1.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPMAAABCCAYAAACCYODvAAAOY0lEQVR4Xu2dBfB9RRXHP6jYhd2JHdiK3djdOuooIrbi2IEjgoEtotiJ3QFiJ4gdiGN3IIqNgTEfZ+/M43rve3v3xruxZ8ZR/7937909u9/dPed8z9kdyJI1kDUwCw3sMIte5E5kDWQNMDUw296zAsfksVucBs4A/AM4fnE9j+zwlMB8auDFwAnAnsC/IvuYfzYPDdwJeBzgf393Hl3qthdTAbOr8lvCIO4F/LNbNeS3TUQDtwX2B24PfG0ibR6smVMAs0B+J/BD4AEZyIPNjbF+6C7AfsCtgG+OtZHbaNfYwbwj8HpgJ8BV+a/bUFL+5ug08AjgocD1wyI/ugZuo0FjBrNtexpwN2BX4FfbUFD+5ig14CL/OuCCwA2Av4yylQM3asxgvjnwJuAOwGED6yV/bvwaOAdwOPAewJ36P+Nvcr8tHCuYHagjAojvnweq30kw4bffDjg42M+LX/A3gfkswLEDD7ZtOgi4JXBF4BcDfz9/bjoaOGlwjl4kmGJ/mE7Tu2/pOjCfAvgw8GXguDWf/gLwQeBhwOXD/z4qBPcvDFwH+Fuwf2N6cGXgU8DewLNiHoj8zcWBZwP3AH4f+cwSfzY1PV0O+AzwjAZzbJbjug7M5wS+A5x2Tc8F6c2AjwNPqFHmq4EHR3qiTwK8F7g0cAXgdy21fk/gosDZgBsBpwzv/m3L987t8SnryTn8WuCmYTP52dwGJ7Y/68B8FeAhwW6tci5cCv5HB31s+JhgPlegW54qLAQSPY6MbUw4KrkrPxXYp8FzdT911XYX/gnwNuDqGcyVqpq6njzNfRp4LvD4DubNJF+xDswG5wXiDyp6JrXy6YFeV8R+BbPhgp+20ITea3d6J5ckkS7lHRnMUeqcop480X0E0ETYBfhNVE9n9qN1YL4x8Eng76U++4zAfRegbVxIWzAbM/xqOAncsQc9T3GS9qCGja+cqp70hUgw2h145cZezvAHm7zZVV0W5OcBXlX64yqYXSlPFrJcYtX2aOCZgXcrfbNrmeok7VoPm943VT2dHfh24GxLJFlcIk5TMJ8p8GK1pcvJDoJZr6IebW1mj+IXCqvloRtmkMDX5vH3lwT6cFBNdZJuAl/Xf5+ynt4P3DDMwaO7VszY39cUzO6cnwAOqeiYYHY1NJz07/B3PeKfBZ4S7Ok6fVwC+Eqwe27Rk9KmPEl7Uknla6espz0CR0He9ouGVNoYvtUEzOcH3g1cF6gKzp87FA0o79g6yrRjJIDoVa4Ss6EODN7zA3pSzJQnaU8qmR2YjbDod/kYcJOlMQebgPlJwNUAOdNNRG61YSH5s8+vedDYsoyvqzYMZTVpRwZznLamrKfTBJLT6QGB3ZanEKexkfwqFsyywT4PfBR4ZEXbfc/JKzzf/lSH2YeAtwJ3rni2GADJKZfpcQCmPEmHnC5T15OJF+Y6yykwEWMxEgtmVzkTwaVsvrBCO/sCjwIkvuuEWBUdEtJC9VBbIaIs8mq/BUgLveaKvd31IEx9knatj7r3TV1PTwyEowcF020ovcV8p1eqbCyY7wW8BrgPID2zLALY47dEE1lfq2LNJv9N7qw1nMoiDU9ut9kvd4/RSOJv+p6kUkWfFxa1Pye2cQyP9a2nvvvohvH24HB13m5bBqPKxoJZp5QrndU+dIKVRe61oaWqukx6tw1laQ9/veJZbWlpeK6o7vB9Sd+T9MzBDDHG2UdorS+9lN/bt5767ofUTpmLRlGu3eNJL7Yfg1FlY8HsAHuErgOziQz3CxTPIixlZw1NaWvLzDF0VSWS5F29rCginbMveV+wo4xj/7qHj8wFzH3rqQfVn+iV5gdYvfPngRY8plJTvS6UsWD2mOxx2aO0R+IqMfQkpU7b+PvBmWXZH3dyj9h1jBwJJbuFrCb5tV2K8UZtcoFmFpbkFLnm2ugmjzwZ+FNHH5wymIfUU0fqrn2NnuxvABaCvFhPC3dqH0YBZtMIjdu9fEMRcllfJkroMPsx4Cq/7shpcrmsMUNeEuSrjuGpihv6uSmDeWhd9fk9Iy9fCnPQeTumGtujAHNfytdpZPEDGWDuoN/r60MDvDeDeQAlR3zC05chqSuF/wjsKjF/wBwD/ztVNCnNn141Lde9a9Zgdid3N7Yiyflapk+mDkhXz2Uwd6XJ9u+RAXa9UOXG/PgqsQCG4PJ0mCqCWO+5x/oYGQzM26huaI0x49fWxRbQOi2aSh/tjvUlrLa1LZj76EdTXY7x9yljEQPmbfR1MDBvo3MCoLiVwJVyzCEdK4ZKmqk7lmkyyHDTWWg5pSoRsCYApCxa2xifqX6zALN+HtmHY5FFgNnVV6fZmMFs+M14ubTVKokBs7cYviRXHO0dW3ln7l3F//8BednazOYxZ5t5CwMw009mMG9hYC1iIGtMT7YplnUpkltoWuNPtrWZG38wP1CpAb3ZnwNkglmUUs5/lXgSNCx6xhZ6/GNgmcXeSDnrY/ZqGOGyDbyCLfTf26MZzL2pttGLNYO+GEhLhjwtJVQl+j7MwW/jzZYIpf8jh6aChk2r9DY/ebTyu7sU0yqlicoKMt5ofW/tc5lsFwjFFFwtu6gXlcHc5cilv+t0YVNwPDzxjenCwV6psilu/0LNEsgt8ePu6mpozNiKnVb0bCJvDICzIqfZLl2JYL1NyPaSsvmKAOjjw/HKetrmWAvmqrTOpu1YOphj0vvMVzczzuOp4hHXcfGyha7EqIOsL8vtetrbdgbbYFTZVDBL2bQaokkSxRFDb66XYMt9blL6p7gJY10lkqYDbVv2AvZfKTxoOpztuncAsBfSvRR4YPAwN/1G+fdLBHOT9L5rhOqrJusUda0FnhuAY9VVIQEXDJ2q2s3XanAEbjv+W38+BcweY6zrZZimTHTQO21cz132l5G9K/JPLd1738hnNv3MY7TF+Fe53pbydcEoysl49NZJ4tE71uZZ990lgjk2vc95oSOqqLe+qkdj86a/ys/v4p5lY8sWnDQDTxNrMZICZj2EKt/SLFXisdVaX7H2b7GSWvVT27kLJpScWxeTVVtYe0XvuXdOdfGNct89DZgd5vUoY0q7G2oyr/PUuht724lH8TJhRnNIJ5W14spValLa7i7/nFDeyjz5xUgKmHcNt05oL5ftY+1nd2Z32B9FarFIWRNg2jiFPRX5eNTPPE0YPnD3Ny0zS/caWAfmN4Rc8iob1lRFuc3OG3Pi28qbAxPPclU6VxcjKWBW+ZYzlRHlFakeuQv6okdmj0vWA4sV22BOs4X/+gpPFdUnvF62jngf295t/G5Ir3xq/+rAXKS5uqCa816+7sjTkuaQBSPaVgbREWuVEXd7w1Kxpl5qn0f1XAqY7YBeYm0Sj5Ym+utMKlhcFsov187e1Glrg+k800nlcaxreXi4rbJMGS1W7z6O3V31YWivfGq768BcgNXogae6E0ofKP7uGJjTbrQhVbyvTK6/Fyq4cHcRckxty+DPpYLZhjowFvkzAVyxwoie4pTKHcXO6RHpri21IBnAhAgnhd5q++gRztDZ6srvVTs6X+RKj1W24ZVP1UUdmA0/eYw23lsF5oI45O88mbUBs45Xw417h2uBU/syyedSwSxgvKXCGl8OgsX+dgxhKcHY5E5mFefqLGvHyetdVW3sZp+34IHljSys7/+3UKDv9jYOxfbr2daGPm7EI7cNr3yqOurAXGTGmcTfN5itHGvpKk29uqIEqf0b/XMpYNYGsqSsR5mi7K6AeVmo7HBscHY0LdeiB1JPpIBrSjxZVbS50U4sgap95mSSmGKbLS4o/9t6YKYqVt09PaZB68Ir7zF9z3BCSembi6w3kmySOjAXDs5jNhyzfX+bndlNRSenmXcuGmXbfFP7J//3FDDrqTY85QRZtTXdmbV9vcZGIGlHNxFjlsYivVvXd7cRd16P/8Yti8vf7et5A2PNi9zHbCfX9T3FK282msSYlLG2He5wHl03SR2Yi9JQPu8iWs71Lmxm/91SP6kgNLpi+NEbVxYVkioGpukACxLjx48JlMiqAZZ1pWNJwDdxhGk7efPFzmGFHvPxd9PE7uvvY/bK14HZOWZKogupO285Bl9wqaV0GtFIFb9vqSBPiRaTXJw0BXORsqg3Wy92lRh+8Mit3dIEzL5LcoGOtLqbMxY3QKUOj9krvy7OLJnGGLLhIo/bq1Jwqd1NdVyliBuAJwjNAX05i5SmYPb3OpY8CtclRciH1asowbypeCQzx9TvyOWtK7/T9L1T/P3UvPLrwGzISX+Fd4lpg6+K/+aJrO7Gk5ix8zSoaeYGclTMA3P8TVMwqwOdC145c+uKGxvl4JrM4AqbWmhAD7RXvFp0P8bxMsdxsU9T88qvS+9znulHMYPJRb7wV/jvnuJ0quqLSfFj6BNwV3Zz8Z7vxUoKmFWWN1DoaPDaGTnVDoYg1xNtOmGb1VHb2VswrDziO7edwratyTEFr3yT9D692sb9jYK44Dv3fN7rgnTQpY7zCwI3QedZ6gayrTHu9LupYLYReq91OOiUkdUjTdKjVBcZSNpWXvwlVdQj1FJlbl5555sebRNqnCdmrMkJSBX9M4Yx3VgOSn3JXJ5rA+a+daANJMVT2/novj+W3z85DcjoOyyQfrzUsKmzdXId3tTgMYPZnV+etnWazFFdYlrhpvFb8t+99G+P4PSSXbZ4GTOYHRxZPa6+RwS+dYqDZPGDPEMFSBAxomKWntGPLC1YQUMqTzqigDYhQkpmlmVrQDKSYTCJSwcvWxUn7v3Yd+aitaZXfiBU8rDuWJZlasCqNNYM2yfUn1umFmp6PRUw23x3aDnCgvnAxJhkHvzpasCoiRVL3JENXWYpaWBKYM6DlzWQNbBGAxnMeXpkDcxEAxnMMxnI3I2sgQzmPAeyBmaigQzmmQxk7kbWQAZzngNZAzPRwH8BUBYscHAb0HQAAAAASUVORK5CYII=)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 75(2x - 10) = 8x2 – 80x
⇒ 150x – 750 = 8x2 – 80x
⇒ 8x2 – 230x + 750 = 0
Now for factorizing the above quadratic equation, two numbers are to be found such that their product is equal to 750 x 8 and their sum is equal to 230
⇒ 8x2 – 200x – 30x +750 = 0
⇒ 8x(x – 25) – 30(x – 25) = 0
⇒ (x – 25 )(8x – 30) = 0
⇒ x = 25, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAiCAMAAAC3OnOQAAAAAXNSR0IArs4c6QAAAD9QTFRFAAAAAAAAAAA6ADqQAGa2OgAAOgA6OpDbZgAAZgA6Zrb/kDoAkDo6kNv/tmYA25A62////7Zm/9uQ//+2///br9x3CAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAg0lEQVQoU62Q2w7DIAxDk5WuFNYr/P+3zg4rY1LVl9UvVpSDUSzyrxZVHRCSo3l+TbI9JpHFyf6EQ/TkAxCCsCCSRuxmh3FW/ZnB96Hy5DiTtbzjv+Tt3d3CNa3ujj/P4y3dWnfsKXnrysTuSldFZd8c/+m+5XP88kd/NYAdNv9dHPkGzgQE7p3qpmcAAAAASUVORK5CYII=)
Time taken by the smaller pipe cannot be
= 3.75 hours.
As in this case, the time taken by the larger pipe will be negative, which is logically not possible.
Therefore, time taken individually by the smaller pipe and the larger pipe will be 25 and 25 − 10 =15 hours respectively.
Question 10.An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains.
Answer:Let the average speed of passenger train be x km/h.
Average speed of express train = (x + 11) km/h
It is given that the time taken by the express train to cover 132 km is 1 hour less than the passenger train to cover the same distance.
As,
![](data:image/png;base64,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)
Time taken by passenger train ![](data:image/png;base64,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)
Time taken by express train ![](data:image/png;base64,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)
Now Time taken by passenger train - Time taken by express train = 1
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 132×11 = x(x+11)
⇒ x2 +11x – 1452 = 0
Now we have to factorize this equation such that the product of numbers is 1452 and their difference is 11
⇒ x2 + 44x – 33x – 1452 = 0
⇒ x(x+44) – 33(x+44) = 0
⇒ (x+44)(x – 33) = 0
⇒ x = - 44, 33
Speed cannot be negative.
Therefore, the speed of the passenger train will be 33 km/h and thus, the speed of the express train will be 33 + 11 = 44 km/h.
Question 11.Sum of the areas of two squares is 468 m2. If the difference of their perimeters is 24 m, find the sides of the two squares.
Answer:Let the sides of the two squares be x m and y m.
Therefore, their perimeter will be 4x and 4y respectively and their areas will be x2 and y2 respectively.
It is given that 4x − 4y = 24 [Difference of perimeter]
or x − y = 6
x = y + 6
Also, x2 + y2 = 468 [sum of squares is 468]
(6+y)2 +y2 = 468
36 + y2 +12y + y2 = 468
2y2 +12y – 432 = 0
y2 + 6y – 216 = 0
y2 +18y – 12y – 216 = 0
y(y+18)(y – 12) = 0
y = - 18 or 12.
However, side of a square cannot be negative.
Hence, the sides of the squares are 12 m and (12 + 6) m = 18 m
Find the roots of the following quadratic equations, if they exist, by the method of completing the square:
(i) 2x2 - 7x + 3 = 0
(ii) 2x2 + x - 4 = 0
(iii)
(iv) 2x2 + x + 4 = 0
Answer:
i) 2x2 – 7x +3 =0
Dividing by coefficient of x2, we get
Adding and subtracting the square of
Use the formula (a-b)2 = a2 + b2 - 2ab to get,
So,
When,
When,
Hence the value of x is 3 and
ii) 2x2 + x – 4 = 0
Dividing by coefficient of
Adding and subtracting the square of
Use the formula (a+b)2 = a2 + b2 + 2ab to get,
iii)
Divide the whole equation by 4 to get,
Add and subtract the square of half of the coefficient of x,
Use the formula (a+b)2 = a2 + b2 + 2ab to get,
∴ Roots of equation are = x =
iv) 2x2 + x + 4
Divide by coefficient of
Adding and subtracting the square of coefficient of x we get,
Use the formula (a+b)2 = a2 + b2 +2ab to get,
Square of a number cannot be negative, Hence roots of the given equation do not exist.
Question 2.
Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.
Answer:
For a quadratic equation: ax2 + bx + c = 0,
The roots of equation is given by formula
i) 2x2 – 7x + 3 = 0
According to the quadratic formula;
∴ The roots are =
=
=
∴ The roots are 3 and
ii) 2x2 + x – 4 = 0
According to the quadratic formula;
=
∴ The roots are =
= =
∴ The roots are =
iii) 4x2 + 4√3x + 3 = 0
According to the quadratic formula;
=
∴ The roots are =
=
∴ The roots are
iv) 2x2 + x + 4 = 0
According to the quadratic formula;
=
∴ The roots are =
=
This is not possible, Hence the roots do not exists.
Question 3.
Find the roots of the following equations:
(i)
(ii)
Answer:
(i)
Taking L.C.M and then cross multiplying we get
According to the quadratic formula, the roots of the equation: a x2 + bx + c = 0 is given by the expression
For the given equation,
So, the roots are,
∴ The roots are =
(ii)
Factorizing the quadratic equation, such that the product of two numbers is 2 and their sum is 3 ( 2 x 1 and 2 + 1)
x(x - 2) - 1(x - 2) = 0
(x - 2)(x - 1) = 0
i.e x - 2 = 0 or x - 1 = 0
therefore,
x = 1 and x = 2 are the roots of the given equation.
Question 4.
The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago and 5 years from now isFind his present age.
Answer:
Let the present age of Rehman be x years.
Three years ago, his age was (x − 3) years.
Five years hence, his age will be (x + 5) years.
It is given that the sum of the reciprocals of Rehman’s ages 3 years ago and 5 years from now is .
∴
⇒
⇒
⇒ 3(2 x + 2) = (x - 3)(x + 5)
⇒ 6 x + 6 = x2 + 2 x – 15
⇒ x2 – 4x - 21 = 0
⇒ x2 – 7 x + 3 x – 21 = 0
⇒ x (x - 7) +3 (x - 7) = 0
⇒ x = 7, -3
However, age cannot be negative.
Therefore, Rehman’s present age is 7 years.
⇒Question 5.
In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.
Answer:
Given : In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of her marks would have been 210.
To find: her marks in two subjects.
Solution:
Let the marks obtained in Mathematics by Shefali be ‘a’.
Given, sum of the marks obtained by Shefali in Mathematics and English is 30.
Marks obtained in english = 30 – a
Also, she got 2 marks more in Mathematics and 3 marks less in English, the product of her marks would have been 210.
⇒(a + 2)(30 – a – 3) = 210
⇒ (a + 2)(27 – a ) = 210
⇒ 27a -a2 + 54 - 2a = 210
⇒-a2 + 25a + 54 = 210
⇒ -a2 + 25a + 54 - 210 = 0
⇒ -a2 + 25a - 156 = 0
⇒a2 – 25a + 156 = 0
In factorization, we write the middle term of the quadratic equation either as a sum of two numbers or difference of two numbers such that the equation can be factorized.
⇒a2 – 13a – 12a + 156 = 0
⇒a(a – 13) – 12(a – 13) = 0
⇒(a – 12)(a – 13) = 0
⇒a = 12 or 13
If marks in mathematics is 12
marks in english is 30 - a = 30 - 12 = 18
If marks in mathematics is 13
marks in english is 30 - a = 30 - 13 = 17
Hence
Marks in Mathematics = 12, Marks in English = 18
Or
Marks in Mathematics = 13, Marks in English = 17
Question 6.
The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.
Answer:
To find: length and breadth of the field
Let the shorter side of the rectangle be x m.
Then, larger side of the rectangle = (x + 30) m
we know,By pythagoras theorem,
(Hypotenuse)2 = (Base)2 + (Perpendicular)2
Diagonal of a rectangle is √[(length)2+ (breadth)2]
Diagonal of the rectangle =
It is given that the diagonal of the rectangle is 60 m more than the shorter side.
∴ = x+60
⇒ x2 + (x+30)2 = (x+60)2
⇒ x2 +x2 +900+60x = x2+3600+120x
⇒ x2 – 60x – 2700 = 0
Now for solving this quadratic equation, we need to factorize 60 in such a way that the product is 2700 and the difference is 60⇒ x(x – 90 )+30(x – 90 ) = 0
⇒ (x - 90)(x+30) = 0
⇒ x = 90, -30
However, side cannot be negative. Therefore, the length of the shorter side will be 90 m.
Hence, length of the larger side will be (90 + 30) m = 120 m
Question 7.
The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.
Answer:
Assumption: Let the larger and smaller number be x and y respectively.
Given:
According to the given question difference of squares of two numbers is 180. and the square of smaller number is 8 times square of the larger number.
So,
⇒ x2 – y2 = 180 eq(i)
⇒ y2 = 8x eq(ii)putting value of eq(ii) in eq(i)
⇒ x2 – 8x = 180
⇒ x2 – 8x – 180 = 0
For factorizing this quadratic equation, the product of numbers should be 180 and their difference should be 8⇒ x2 – 18x +10x – 180 = 0
⇒ x(x – 18 ) +10(x – 18 ) = 0
⇒ (x – 18 )(x+10) = 0
⇒ x = 18, -10
The larger number cannot be negative as it makes the square of smaller number negative, which is not possible.
Therefore the larger number will be 18 only.
x = 18
∴ y2 = 8x
= 8×18
= 144
=
∴ smaller number =
Therefore, the numbers are 18 and 12 or 18 and −12.
Note: As we need to solve this problem using one variable, so the variable y is written in terms of variable x.
Question 8.
A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.
Answer:
To find: Speed of the train
Let the speed of the train be x km/hr.
Time taken to cover 360 km = hr, As
Now, given that if the speed would be 5 km/hr more, the same distance would be covered in 1 hour less, i.e.
if speed = x + 5, and
Now we can form the quadratic equation from this equation
Now, cross multiplying we get
⇒ 360 x - x2 + 1800 - 5 x = 360 x
⇒ x2 + 5 x – 1800 = 0
Now we have to factorize in such a way that the product of the two numbers is 1800 and the difference is 5
⇒ x2 + 45 x – 40 x – 1800 = 0
⇒ x(x + 45) – 40(x+45) = 0
⇒ (x+45)(x – 40 ) = 0
⇒ x = - 45, 40
since, the speed of train can’t be negative,
so, speed will be 40 km/hour.
Question 9.
Two water taps together can fill a tank in hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
Answer:
Let the time taken by the smaller pipe to fill the tank be x hr.
Time taken by the larger pipe = (x − 10) hr
Part of the tank filled by a smaller pipe in 1 hour =
Part of the tank filled by the larger pipe in 1 hour =
It is given that the tank can be filled in hours by both the pipes together.
So 75/8 hours, multiplied by the sum of parts filled with both pipes in one hour equal to complete work i.e 1.
⇒
⇒
⇒
⇒ 75(2x - 10) = 8x2 – 80x
⇒ 150x – 750 = 8x2 – 80x
⇒ 8x2 – 230x + 750 = 0
Now for factorizing the above quadratic equation, two numbers are to be found such that their product is equal to 750 x 8 and their sum is equal to 230⇒ 8x2 – 200x – 30x +750 = 0
⇒ 8x(x – 25) – 30(x – 25) = 0
⇒ (x – 25 )(8x – 30) = 0
⇒ x = 25,
Time taken by the smaller pipe cannot be = 3.75 hours.
As in this case, the time taken by the larger pipe will be negative, which is logically not possible.
Therefore, time taken individually by the smaller pipe and the larger pipe will be 25 and 25 − 10 =15 hours respectively.
Question 10.
An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains.
Answer:
Let the average speed of passenger train be x km/h.
Average speed of express train = (x + 11) km/h
It is given that the time taken by the express train to cover 132 km is 1 hour less than the passenger train to cover the same distance.
As,Time taken by passenger train
Time taken by express train
Now Time taken by passenger train - Time taken by express train = 1
⇒
⇒ 132×11 = x(x+11)
⇒ x2 +11x – 1452 = 0
Now we have to factorize this equation such that the product of numbers is 1452 and their difference is 11⇒ x2 + 44x – 33x – 1452 = 0
⇒ x(x+44) – 33(x+44) = 0
⇒ (x+44)(x – 33) = 0
⇒ x = - 44, 33
Speed cannot be negative.
Therefore, the speed of the passenger train will be 33 km/h and thus, the speed of the express train will be 33 + 11 = 44 km/h.
Question 11.
Sum of the areas of two squares is 468 m2. If the difference of their perimeters is 24 m, find the sides of the two squares.
Answer:
Let the sides of the two squares be x m and y m.
Therefore, their perimeter will be 4x and 4y respectively and their areas will be x2 and y2 respectively.
It is given that 4x − 4y = 24 [Difference of perimeter]
or x − y = 6
x = y + 6
Also, x2 + y2 = 468 [sum of squares is 468]
(6+y)2 +y2 = 468
36 + y2 +12y + y2 = 468
2y2 +12y – 432 = 0
y2 + 6y – 216 = 0
y2 +18y – 12y – 216 = 0
y(y+18)(y – 12) = 0
y = - 18 or 12.
However, side of a square cannot be negative.
Hence, the sides of the squares are 12 m and (12 + 6) m = 18 m
Exercise 4.4
Question 1.Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:
(i) 2x2 – 3x + 5 = 0
(ii) ![](data:image/png;base64,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)
(iii) 2x2 – 6x + 3 = 0
Answer:There are three types of roots that are possible for a quadratic equation:
For a quadratic equation: a x2 + b x + c = 0, we know that its roots is given by the formula
![](data:image/png;base64,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)
(i) Comparing this equation with ax2 + bx + c = 0, we obtain,
a = 2, b = −3, c = 5
Discriminant D = b2 − 4ac = (− 3)2 − 4 (2) (5) = 9 − 40 = −31
As, b2 − 4ac < 0,
Therefore, no real root is possible for the given equation.
(ii) Comparing this equation with ax2 + bx + c = 0,
we obtain,
a = 3
b = -4√3
c = 4
![](data:image/png;base64,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)
Discriminant = 48 − 48 = 0
As, b2 − 4ac = 0,
Therefore, real roots exist for the given equation and they are equal to each other.
And the roots will be
and
.
![](data:image/png;base64,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)
Therefore, the roots are
and
.
(iii) Comparing this equation with ax2+bx + c = 0,
we obtain,
a = 2, b = −6, c = 3
Discriminant = b2 − 4ac = (− 6)2 − 4 (2) (3) = 36 − 24 = 12
As, b2 − 4ac > 0,
Therefore, distinct real roots exist for this equation as follows.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
So, the roots are
or
.
Question 2.Find the values of k for each of the following quadratic equations, so that they have two equal roots.
(i) 2x2+ kx + 3 = 0
(ii) kx (x – 2) + 6 = 0
Answer:We know that if an equation ax2 + bx + c = 0 has two equal roots,
its discriminant
(b2 − 4ac) will be 0.
(i) 2x2 + kx + 3 = 0
Comparing equation with ax2 + bx + c = 0, we obtain,
a = 2, b = k, c = 3
Discriminate = b2 − 4ac = (k)2− 4(2) (3) = k2 − 24
For equal roots,
Discriminant = 0
k2 − 24 = 0
k2 = 24
=![](data:image/png;base64,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)
(ii) kx (x − 2) + 6 = 0
or kx2− 2kx + 6 = 0
Comparing this equation with ax2 + bx + c = 0, we obtain,
a = k, b = −2 k, c = 6
Discriminant = b2 − 4ac = (− 2k)2 − 4 (k) (6) = 4k2 − 24k
For equal roots, b2 − 4ac = 0
= 4k2 − 24k = 0
= 4k (k − 6) = 0
Either 4k = 0 or k = 6
= k = 0 or k = 6
However, if k = 0, then the equation will not have the terms ‘x2’ and ‘x’.
Therefore, if this equation has two equal roots, k should be 6 only.
Question 3.Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2? If so, find its length and breadth.
Answer:Let the breadth of mango grove be l.
Length of mango grove will be 2l.
Area of mango grove = (2l) (l) = 2l2
2l2 = 800
=
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGMAAAAZCAMAAAASXGAAAAAAAXNSR0IArs4c6QAAAEhQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6OpDbZgAAZgA6ZgBmZmZmZrb/kDoAkNv/tmYAtv//25A62////7Zm/9uQ//+2///b4uXVgwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABI0lEQVRIS+1UyxKDIAyU1j5sK1JE+P8/bR4g9BH04kwP5tDJjBs2u0naNHv8pQO+U+q0bWf+NjTuMGxFYtTxSW9Pl804go4e2WtNhmWVQavYU86EsgIQNL/tquOYzsQRdDsGjcJzJlIkKFh07hFlqxS+exAHzcwpqMiZwFECHI0DVTgi+xnmyhthEOw7AOdMKpmhIKAdsUxBiByuHYkDDUIwupCyilUMxbLls5g3m1/mUaSMOQz2SBEvoAT47nOdfsABAjpsL3N8qSk5VlyFSy1GDpxH8ko0oQTYeIFLJ/42c9AVZ55M+BJfAgz6uiKYw9La4k/OhOIMgJH7O/+X1IPv/H2l2A8hMhT6WWUWzCSSKMUP47pXPVgELMnav+8OVBx4AcaqE1vraio8AAAAAElFTkSuQmCC)
⇒ l2 = 400
![](data:image/png;base64,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)
However, length cannot be negative.
Therefore, breadth of mango grove = 20 m
Length of mango grove = 2 × 20 = 40 m
Question 4.Is the following situation possible? If so, determine their present ages.
The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.
Answer:Let the age of one friend be x years.
Age of the other friend will be (20 − x) years.
4 years ago,
age of 1st friend = (x − 4) years
And, age of 2nd friend = (20 − x − 4) = (16 − x) years
Given that,
(x − 4) (16 − x) = 48
16x − 64 − x2 + 4x = 48
x2 − 20x + 112 = 0
Comparing this equation with ax2 + bx + c = 0, we obtain
a = 1, b = −20, c = 112
Discriminant = b2 − 4ac = (− 20)2 − 4 (1) (112) = 400 − 448 = −48
As b2 − 4ac < 0,
Therefore, no real root is possible for this equation and hence, this situation is not possible.
Question 5.Is it possible to design a rectangular park of perimeter 80 m and area 400 m2? If so, find its length and breadth.
Answer:Let the length and breadth of the park be l and b.
Perimeter = 2 (l + b) = 80
l + b = 40 Or, b = 40 – l
Area = l × b = l (40 − l)
= 40l − l2= 400 Given
l2 − 40l + 400 = 0
Comparing this equation with al2 + bl + c = 0, we obtain
a = 1, b = −40, c = 400
Discriminant D = b2 − 4ac = (− 40)2 −4 (1) (400) = 1600 − 1600 = 0
As b2 − 4ac = 0,
Therefore, this equation has equal real roots and hence, this situation is possible.
Root of this equation,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEsAAAAsCAMAAAAaYbU6AAAAAXNSR0IArs4c6QAAAFRQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6OjqQOpCQOpDbZgAAZgA6ZgBmZrb/kDoAkLaQkNv/tmYAtv//25A627Zm2////7Zm/9uQ//+2///bW7PD4wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABKUlEQVRIS+1W0RKCIBAULc20JJEC/P//7DiwMacxR7a37kF90OVu2V3Jsn+tM6BEMcA4UiUMapQNDMtdehiWKQYYZfooO9diOFN5n43y8AAM6lqiHoRlT0Q9aEbtlWpEBxgxU54pviTXKKkvTfQDytZ3gTTkoiWYcgl3lBjh+haRZrcVRCDMHJkdsKkBQkPExlAz6pWYaqcA2eyr9WGJ16KzB8Jgs4OKzR4qeUaMzyfq3RUlCv1Dq2/cBkM7uFM5yxX8H8i1GMcoL0qNifrg5IBF4+a3OjH1Fc/ouzMiMQNDWNmKxk3NrZih3FyiEWO+hNvMiBs19fZaPBLGM0USXZygpiGREV1afMuttW4DQaQyP6M5t43dL4oYT9QOqauktMIF/R7CV795AoujDVktdgfHAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
Therefore, length of park, l = 20 m
And breadth of park, b = 40 − l = 40 − 20 = 20 m
Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:
(i) 2x2 – 3x + 5 = 0
(ii)
(iii) 2x2 – 6x + 3 = 0
Answer:
There are three types of roots that are possible for a quadratic equation:
For a quadratic equation: a x2 + b x + c = 0, we know that its roots is given by the formula
(i) Comparing this equation with ax2 + bx + c = 0, we obtain,
a = 2, b = −3, c = 5
Discriminant D = b2 − 4ac = (− 3)2 − 4 (2) (5) = 9 − 40 = −31
As, b2 − 4ac < 0,
Therefore, no real root is possible for the given equation.
(ii) Comparing this equation with ax2 + bx + c = 0,
we obtain,
a = 3
b = -4√3
c = 4
Discriminant = 48 − 48 = 0
As, b2 − 4ac = 0,
Therefore, real roots exist for the given equation and they are equal to each other.
And the roots will be and
.
Therefore, the roots are and
.
(iii) Comparing this equation with ax2+bx + c = 0,
we obtain,
a = 2, b = −6, c = 3
Discriminant = b2 − 4ac = (− 6)2 − 4 (2) (3) = 36 − 24 = 12
As, b2 − 4ac > 0,
Therefore, distinct real roots exist for this equation as follows.
So, the roots are or
.
Question 2.
Find the values of k for each of the following quadratic equations, so that they have two equal roots.
(i) 2x2+ kx + 3 = 0
(ii) kx (x – 2) + 6 = 0
Answer:
We know that if an equation ax2 + bx + c = 0 has two equal roots,
its discriminant
(b2 − 4ac) will be 0.
(i) 2x2 + kx + 3 = 0
Comparing equation with ax2 + bx + c = 0, we obtain,
a = 2, b = k, c = 3
Discriminate = b2 − 4ac = (k)2− 4(2) (3) = k2 − 24
For equal roots,
Discriminant = 0
k2 − 24 = 0
k2 = 24
=
(ii) kx (x − 2) + 6 = 0
or kx2− 2kx + 6 = 0
Comparing this equation with ax2 + bx + c = 0, we obtain,
a = k, b = −2 k, c = 6
Discriminant = b2 − 4ac = (− 2k)2 − 4 (k) (6) = 4k2 − 24k
For equal roots, b2 − 4ac = 0
= 4k2 − 24k = 0
= 4k (k − 6) = 0
Either 4k = 0 or k = 6
= k = 0 or k = 6
However, if k = 0, then the equation will not have the terms ‘x2’ and ‘x’.
Therefore, if this equation has two equal roots, k should be 6 only.
Question 3.
Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2? If so, find its length and breadth.
Answer:
Let the breadth of mango grove be l.
Length of mango grove will be 2l.
Area of mango grove = (2l) (l) = 2l2
2l2 = 800
=
⇒
⇒ l2 = 400
However, length cannot be negative.
Therefore, breadth of mango grove = 20 m
Length of mango grove = 2 × 20 = 40 m
Question 4.
Is the following situation possible? If so, determine their present ages.
The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.
Answer:
Let the age of one friend be x years.
Age of the other friend will be (20 − x) years.
4 years ago,
age of 1st friend = (x − 4) years
And, age of 2nd friend = (20 − x − 4) = (16 − x) years
Given that,
(x − 4) (16 − x) = 48
16x − 64 − x2 + 4x = 48
x2 − 20x + 112 = 0
Comparing this equation with ax2 + bx + c = 0, we obtain
a = 1, b = −20, c = 112
Discriminant = b2 − 4ac = (− 20)2 − 4 (1) (112) = 400 − 448 = −48
As b2 − 4ac < 0,
Therefore, no real root is possible for this equation and hence, this situation is not possible.
Question 5.
Is it possible to design a rectangular park of perimeter 80 m and area 400 m2? If so, find its length and breadth.
Answer:
Let the length and breadth of the park be l and b.
Perimeter = 2 (l + b) = 80
l + b = 40 Or, b = 40 – l
Area = l × b = l (40 − l)
= 40l − l2= 400 Given
l2 − 40l + 400 = 0
Comparing this equation with al2 + bl + c = 0, we obtain
a = 1, b = −40, c = 400
Discriminant D = b2 − 4ac = (− 40)2 −4 (1) (400) = 1600 − 1600 = 0
As b2 − 4ac = 0,
Therefore, this equation has equal real roots and hence, this situation is possible.
Root of this equation,
Therefore, length of park, l = 20 m
And breadth of park, b = 40 − l = 40 − 20 = 20 m