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Solving a Quadratic Equation by Factorization Method

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Question

Solve the following quadratic equations by factorization:

$\frac{1}{2 a + b + 2 x} = \frac{1}{2 a} + \frac{1}{b} + \frac{1}{2 x}$

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Solution

$\frac{1}{2 a + b + 2 x} = \frac{1}{2 a} + \frac{1}{b} + \frac{1}{2 x} \Rightarrow \frac{1}{2 a + b + 2 x} - \frac{1}{2 a} = \frac{1}{b} + \frac{1}{2 x} \Rightarrow \frac{2 a - (2 a + b + 2 x)}{(2 a + b + 2 x) (2 a)} = \frac{2 x + b}{2 b x} \Rightarrow \frac{- b - 2 x}{4 a^{2} + 2 a b + 4 a x} = \frac{2 x + b}{2 b x} \Rightarrow \frac{- 1 (2 x + b)}{4 a^{2} + 2 a b + 4 a x} = \frac{2 x + b}{2 b x} \Rightarrow - 2 b x (2 x + b) = (4 a^{2} + 2 a b + 4 a x) (2 x + b) \Rightarrow (4 a^{2} + 2 a b + 4 a x) (2 x + b) + 2 b x (2 x + b) = 0 \Rightarrow (2 x + b) (4 a^{2} + 2 a b + 4 a x + 2 b x) = 0 \Rightarrow 2 x + b = 0 or 4 a^{2} + 2 a b + (4 a + 2 b) x = 0 \Rightarrow x = - \frac{b}{2} or x = - \frac{4 a^{2} + 2 a b}{4 a + 2 b} \Rightarrow x = - \frac{b}{2} or x = - \frac{a (4 a + 2 b)}{4 a + 2 b} \Rightarrow x = - \frac{b}{2} or x = - a$

Hence, the factors are $- a$ and $- \frac{b}{2}$ .

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