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If the equation $x^{4} + 4 x^{3} - 2 x^{2} - 12 x + 9 = 0$ has two pairs of equal roots, find the roots of the equation.

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-3, −3, 1, 1

1, −1, 1, 1

4, −2, -1, 1

2, 2, 1, -1

answer is A.

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Detailed Solution

Given equation is,

x^{4} + 4 x^{3} - 2 x^{2} - 12 x + 9 = 0

Here, two pairs of equal roots of the given equation.
Suppose roots = a, a, b, b
So, sum of the roots,

a + a + b + b = \frac{coefficient of x}{coefficient of x^{3}}

2 a + 2 b = - \frac{12}{4}

2 a + 2 b = - 4

\Rightarrow a + b = - 2

Product of the roots,

a^{2} b^{2} = \frac{(constant term))}{coefficient of x^{4}}

a^{2} b^{2} = \frac{9}{1} = 9

ab = \pm 3

In the first case,

ab = 3

(a - b)^{2} = (a + b)^{2} - 4 ab

(a - b)^{2} = 4 - 12 = - 8

This is not possible case. So, ab

= 3

.
In the second case,

ab = - 3

Thus,

(a - b)^{2} = (a + b)^{2} - 4 ab

(a - b)^{2} = 4 - 4 (- 3) = 4 + 12 = 16

(a - b) = \pm 4

a + b = - 2

a - b = 4 or

a - b = - 4

So,

\Rightarrow 2 a = 2 or 2 a = - 6

\Rightarrow a = 1 or a = - 3

\Rightarrow b = - 3 or b = 1

So, the roots of the equation are−3,−3, 1, 1.
Correct option is 1.

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