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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

If the roots of the equation (a^{2 }+ b^{2})x^{2 }− 2b(a + c)x + (b^{2}+c^{2}) = 0 are equal then

Solution:

(a^{2 }+ b^{2})x^{2 }− 2b(a + c)x + (b^{2}+c^{2}) = 0

Roots are real and equal ∴ D = 0

D = b^{2 }− 4ac = 0

⇒ [−2b(a+c)]^{2 }− 4(a^{2 }+ b^{2})(b^{2 }+ c^{2}) = 0

⇒ b^{2}(a^{2 }+ c^{2 }+ 2ac) −(a^{2}b^{2} + a^{2}c^{2} + b^{4} + c^{2}c^{2}) = 0

⇒ b^{2}a^{2 }+ b^{2}c^{2 }+ 2acb^{2 }− a^{2}b^{2 }− a^{2}c^{2 }− b^{4} − b^{2}c^{2} = 0

⇒ 2acb^{2 }− a^{2}c^{2 }− 2acb^{2 }= 0

⇒ (b^{2 }− ac)^{2 }= 0

⇒ b^{2} = ac

QUESTION: 3

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

Solution:

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:

__Any quadratic equation is of the form:__ **x ^{2} - (sum of the roots)x + (product of the roots) = 0**

where x is a real variable.

As the sum of the roots is 13 and the 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:

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:

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

__ Sum of Roots:__ 2a + 3a = 5a = -(- 5/2) = 5/2

⇒ a = 1/2

⇒ b = 12a

QUESTION: 7

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

Solution:

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.

x = 9 [∵ x is positive]

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:

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

Sum of roots = -(-12) = 12

⇒ x + 3x = 4x = 12

⇒ x = 3

__ Product of the roots:__ 3x

QUESTION: 9

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

Solution:

Let 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

⇒ x = 11 [∵ The numbers are positive, **i.e. **2x > 0 ⇒ x > 0]

Thus, Required numbers are **20, 22, 24**.

QUESTION: 10

For all x, x^{2 }+ 2ax + (10 − 3a) > 0, then the interval in which a lies, is?

Solution:

In f(x) = ax^{2} + bx + c

When a > 0 and D < 0

Then f(x) is always positive.

x^{2} + 2ax + 10 − 3a > 0, ∀x ∈ R

⇒ D < 0

⇒ 4a^{2} − 4(10 − 3a) < 0

⇒ a^{2} + 3a − 10 < 0

⇒ (a+5)(a−2) < 0

⇒ a ∈ (−5,2)

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