If (x + a) is a factor of $x^{2} + p x + q$ and $x^{2} + m x + n$ , then find the value of a.

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Solution

Let f(x) = $x^{2} + p x + q$
And q(x) = $x^{2} + m x + n$
(x + a) is a factor of f(x) and g(x)
By remainder theorem,
$f (- a) = {(- a)}^{2} + p (- a) + q = 0$
$\Rightarrow a^{2} - a p + q = 0$ ...(i)
Also,
$g (- a) = {(- a)}^{2} + m (- a) + n = 0$
$\Rightarrow a^{2} - a m + n = 0$ ...(ii)
Subtracting (ii) from (i)
-a( p - m) + q - n = 0
$\Rightarrow - a (p - m) = - (q - n) \Rightarrow a (p - m) = q - n \Rightarrow a = \frac{n - q}{m - p}$

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