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

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Solution

$\frac{a}{x - b} + \frac{b}{x - a} = 2$
$\Rightarrow \frac{a x - a^{2} + b x - b^{2}}{(x - a) (x - b)} = 2 \Rightarrow a x - a^{2} + b x - b^{2} = 2 x^{2} - 2 b x - 2 a x + 2 a b \Rightarrow 2 x^{2} - 2 b x - 2 a x + 2 a b - a x + a^{2} - b x + b^{2} = 0 \Rightarrow 2 x^{2} + x [- 2 b - 2 a - a - b] + a^{2} + b^{2} + 2 a b = 0 \Rightarrow 2 x^{2} - 3 x [a + b] + {(a + b)}^{2} = 0$
$\Rightarrow 2 x^{2} - 2 (a + b) x - (a + b) x + {(a + b)}^{2} = 0 \Rightarrow 2 x [x - (a + b)] - (a + b) [x - (a + b)] = 0 \Rightarrow [2 x - (a + b)] [x - (a + b)] = 0$
So, the value of x will be
$x = \frac{a + b}{2}, a + b$

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