Determine $p$ and $q$ if $p q = 32$ and $p + q = - 12$

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Solution

If $p + q$ and $p q$ are known then quadratic equation corresponding to roots as $p$ and $q$ is given by,
$x^{2} - (p + q) x + p q = 0$
$\Rightarrow x^{2} + 12 x + 32 = 0$ , (substitute the given values)
$\Rightarrow x^{2} + 4 x + 8 x + 32 = 0$ , (split the middle term)
$\Rightarrow (x^{2} + 4 x) + (8 x + 32) = 0$ , (group pair of terms)
$\Rightarrow x (x + 4) + 8 (x + 4) = 0$ , (factor each binomials)
$\Rightarrow (x + 4) (x + 8) = 0$ , (factor out common factor)
$\Rightarrow x = - 4$ or $x = - 8$ , (set each factor to $0$ )
Hence $p$ and $q$ are $- 8$ and $- 4$

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Determine p and q if pq=32 and p+q=−12

Determine $p$ and $q$ if $p q = 32$ and $p + q = - 12$