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QUESTION: 1

Find the roots of the quadratic equation: x^{2} + 2x - 15 = 0?

Solution:

x^{2} + 5x - 3x - 15 = 0

x(x + 5) - 3(x + 5) = 0

(x - 3)(x + 5) = 0

⇒ x = 3 or x = -5.

QUESTION: 2

Find the roots of the quadratic equation: 2x^{2} + 3x - 9 = 0?

Solution:

2x^{2} + 6x - 3x - 9 = 0

2x(x + 3) - 3(x + 3) = 0

(x + 3)(2x - 3) = 0

⇒ x = -3 or x = 3/2.

QUESTION: 3

The roots of the equation 3x^{2} - 12x + 10 = 0 are?

Solution:

Explanation:

The discriminant of the quadratic equation is (-12)^{2} - 4(3)(10) i.e., 24. As this is positive but not a perfect square, the roots are irrational and unequal.

QUESTION: 4

If the roots of a quadratic equation are 20 and -7, then find the equation?

Solution:

Explanation:

Any quadratic equation is of the form

x^{2} - (sum of the roots)x + (product of the roots) = 0 ---- (1)

where x is a real variable. As sum of the roots is 13 and product of the roots is -140, the quadratic equation with roots as 20 and -7 is: x^{2} - 13x - 140 = 0.

QUESTION: 5

The sum and the product of the roots of the quadratic equation x^{2} + 20x + 3 = 0 are?

Solution:

Explanation:

Sum of the roots and the product of the roots are -20 and 3 respectively.

QUESTION: 6

If the roots of the equation 2x^{2} - 5x + b = 0 are in the ratio of 2:3, then find the value of b?

Solution:

Explanation:

Let the roots of the equation 2a and 3a respectively.

2a + 3a = 5a = -(- 5/2) = 5/2 => a = 1/2

Product of the roots: 6a^{2} = b/2 => b = 12a^{2}

a = 1/2, b = 3.

QUESTION: 7

The sum of the squares of two consecutive positive integers exceeds their product by 91. Find the integers?

Solution:

Explanation:

Let the two consecutive positive integers be x and x + 1

x^{2} + (x + 1)^{2} - x(x + 1) = 91

x^{2} + x - 90 = 0

(x + 10)(x - 9) = 0 => x = -10 or 9.

As x is positive x = 9

Hence the two consecutive positive integers are 9 and 10.

QUESTION: 8

One root of the quadratic equation x^{2} - 12x + a = 0, is thrice the other. Find the value of a?

Solution:

Explanation:

Let the roots of the quadratic equation be x and 3x.

Sum of roots = -(-12) = 12

a + 3a = 4a = 12 => a = 3

Product of the roots = 3a^{2} = 3(3)^{2} = 27.

QUESTION: 9

The sum of the square of the three consecutive even natural numbers is 1460. Find the numbers?

Solution:

Explanation:

Three consecutive even natural numbers be 2x - 2, 2x and 2x + 2.

(2x - 2)^{2} + (2x)^{2} + (2x + 2)^{2} = 1460

4x^{2} - 8x + 4 + 4x^{2} + 8x + 4 = 1460

12x^{2} = 1452 => x^{2} = 121 => x = ± 11

As the numbers are positive, 2x > 0. Hence x > 0. Hence x = 11.

Required numbers are 20, 22, 24.

QUESTION: 10

If a and b are the roots of the equation x^{2} - 9x + 20 = 0, find the value of a^{2} + b^{2} + ab?

Solution:

Explanation:

a^{2} + b^{2} + ab = a^{2} + b^{2} + 2ab - ab

i.e., (a + b)^{2} - ab

from x^{2} - 9x + 20 = 0, we have

a + b = 9 and ab = 20. Hence the value of required expression (9)^{2} - 20 = 61.

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