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Solving a Quadratic Equation by Factorization Method

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Question

Solve the following quadratic equation by factorization.
$\frac{a}{x - a} + \frac{b}{x - b} = \frac{2 c}{x - c}$

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Solution

$fraction numerator a left parenthesis x minus b right parenthesis plus b left parenthesis x minus a right parenthesis over denominator left parenthesis x minus b right parenthesis left parenthesis x minus a right parenthesis end fraction equals fraction numerator 2 c over denominator x minus c end fraction fraction numerator a x minus a b plus b x minus a b over denominator left parenthesis x squared minus b x minus a x plus a b right parenthesis end fraction equals fraction numerator 2 c over denominator x minus c end fraction$

= $(x - c) (a x - 2 a b + b x) = 2 c (x^{2} - b x - a x + a b)$

= $(a + b) x^{2} - 2 a b x - (a + b) c x + 2 a b c = 2 c x^{2} - 2 c (a + b) x + 2 a b c$
= $(a + b - 2 c) x^{2} - 2 a b x + a c x + b c x = 0$
= $x ((a + b - 2 c) x - 2 a b + a c + b c) = 0$

$x = \frac{2 a b - a c - b c}{a + b - 2 c}$

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