Solve for x- 1-2 a-b-2 x-1-2 a-1-b-1-2 x

1/2a+b+2x=1/2a+1/b+1/2x ⇒1/2a+b+2x 1/2x=1/2a+1/b ⇒2x 2a b 2x/(2a+b+2x)2x=2a+b/2ab ⇒ (2a+b)/(2a+b+2x)2x=2a+b/2ab ⇒ 1/(2a+b+2x)2x=1/2ab ⇒ ab= x(2a+b+2x) ⇒ ab+ x(2 ...

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Standard X

Mathematics

Componendo and Dividendo

Solve for x. ...

Question

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

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Solution

$\frac{1}{2 a + b + 2 x} = \frac{1}{2 a} + \frac{1}{b} + \frac{1}{2 x}$
$\Rightarrow \frac{1}{2 a + b + 2 x} - \frac{1}{2 x} = \frac{1}{2 a} + \frac{1}{b}$
$\Rightarrow \frac{2 x - 2 a - b - 2 x}{(2 a + b + 2 x) 2 x} = \frac{2 a + b}{2 a b}$
$\Rightarrow \frac{- (2 a + b)}{(2 a + b + 2 x) 2 x} = \frac{2 a + b}{2 a b}$
$\Rightarrow \frac{- 1}{(2 a + b + 2 x) 2 x} = \frac{1}{2 a b}$
$\Rightarrow a b = - x (2 a + b + 2 x)$
$\Rightarrow a b + x (2 a + b) + 2 x^{2} = 0$
$\Rightarrow 2 x^{2} + x (2 a + b) + a b = 0$
$\Rightarrow x = \frac{- (2 a + b) \pm \sqrt{(2 a + b)^{2} - 4 \times 2 \times a b}}{4}$
$\Rightarrow x = \frac{- (2 a + b) \pm \sqrt{(2 a + b)^{2} - 8 a b}}{4}$
$\Rightarrow x = \frac{- (2 a + b) \pm \sqrt{4 a^{2} + 4 a b + b^{2} - 8 a b}}{4}$
$\Rightarrow x = \frac{- (2 a + b) \pm \sqrt{4 a^{2} - 4 a b + b^{2}}}{4}$
$\Rightarrow x = \frac{- (2 a + b) \pm \sqrt{(2 a - b)^{2}}}{4}$
$\Rightarrow x = \frac{- (2 a + b) \pm (2 a - b)}{4}$
$\Rightarrow x = \frac{- 2 a - b + 2 a - b}{4} o r x = \frac{- 2 a - b - 2 a + b}{4}$
$\Rightarrow x = \frac{- 2 b}{4} o r x = \frac{- 4 a}{4}$
$∴ x = \frac{- b}{2}, - a$

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