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Rational Expression

Solve the equ...

Question

Solve the equation: $2 (\frac{2 x - 1}{x + 3}) - 3 (\frac{x + 3}{2 x - 1}) = 5, (x \neq - 3, \frac{1}{2})$

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Solution

$\frac{2 (2 x - 1)^{2} - 3 (x + 3)^{2}}{(x + 3) (2 x - 1)} = 5$
$\Rightarrow 2 (4 x^{2} + 1 - 4 x) - 3 (x^{2} + 9 + 6 x)$
$= 5 (2 x^{2} + 5 x - 3)$
$\Rightarrow 5 x^{2} - 26 x - 25 = 10 x^{2} + 25 x - 15$
$\Rightarrow 5 x^{2} + 51 x + 10 = 0$
$\Rightarrow 5 x^{2} + 50 x + x + 10 = 0$
$\Rightarrow 5 x + (x + 10) + 1 (x + 10) = 0$
$\Rightarrow (5 x + 1) (x + 10) = 0$
$\Rightarrow x = \frac{- 1}{5}; x = - 10$

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