If x + a is a common factor of expressions $f (x) = x^{2} + p x + q a n d g (x) = x^{2} + m x + n;$ show that : $a = \frac{n - q}{m - p}$

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Solution

Since x+a is the factor of both the equations so we can say
F(-a)=0
and
G(-a)=0

${(- a)}^{2}$ + p·(-a) + q = 0 --> $a^{2}$ -ap + q = 0
${(- a)}^{2}$ + m·(-a) + n = 0 --> $a^{2}$ -am + n = 0
Since they both equal 0, you can set them equal to each other:
$a^{2}$ -ap + q = $a^{2}$ -am + n
-ap + q + -am + n
am - ap = n-q
a(m-p) = n-q

$a equals fraction numerator n minus q over denominator m minus p end fraction$

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