If the roots of the quadratic equation $k x^{2} + (a + b) x + a b = 0$ are $(- 1, - b)$ , find the value of k.

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Solution

$\begin{matrix} R o o t s a r e (- 1, - b), t h e n k = ? T h e n, s a t i s f y t h e r o o t s i n e q u a t i o n \Rightarrow k {(- 1)}^{2} + (a + b) (- 1) + a b = 0 \Rightarrow k - (a + b) + a b = 0 \Rightarrow k = (a + b) - a b . . . . (1) a g a i n p u t x = - b \Rightarrow k {(- b)}^{2} + (a + b) (- b) + a b = 0 \Rightarrow k b^{2} - b^{2} - a b + a b = 0 \Rightarrow k b^{2} - b^{2} = 0 \Rightarrow k b^{2} = b^{2} k = 1 \end{matrix}$

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