Short-Answer Questions

Solve: $x^{2} + 5 x - (a^{2} + a - 6) = 0$ [CBSE 2015]

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Solution

$x^{2} + 5 x - (a^{2} + a - 6) = 0 \Rightarrow x^{2} + 5 x - (a + 3) (a - 2) = 0 \Rightarrow x^{2} + [(a + 3) - (a - 2)] x - (a + 3) (a - 2) = 0 \Rightarrow x^{2} + (a + 3) x - (a - 2) x - (a + 3) (a - 2) = 0$
$\Rightarrow x [x + (a + 3)] - (a - 2) [x + (a + 3)] = 0 \Rightarrow [x + (a + 3)] [x - (a - 2)] = 0 \Rightarrow x + (a + 3) = 0 or x - (a - 2) = 0 \Rightarrow x = - (a + 3) or x = (a - 2)$

Hence, $- (a + 3)$ and $(a - 2)$ are the roots of the given equation.

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