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Question
Solve the following equation by factorization
`(1)/(2a + b + 2x) = (1)/(2a) + (1)/b + (1)/(2x)`
Solution
`(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) = (b + 2a)/(2ab)`
⇒ `(-(2a + b))/((2a + b + 2x)2x) = ((2a + b))/(2ab)`
⇒ `(-1)/((2a + b + 2x)2x) = (1)/(2ab)`
⇒ -2ab = (2a + b + 2x)2x
⇒ 4ax + 2xb + 4x2 = -2ab
⇒ 4x2 + 2bx + 4ax + 2ab = 0
⇒ 2x(2x + b) + 2a(2x + b) = 0
⇒ (2x + 2a)(2x + b) = 0
⇒ 2x + 2a = 0 or 2x + b = 0
`x = -a or x = -b/(2)`
Hence, the roots of the given equation are
`-a and -b/(2)`.
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