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Reducing Equations to Simpler Form

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Question

Solve for x
$\frac{\sqrt{4 x + 1} + \sqrt{x + 3}}{\sqrt{4 x + 1} - \sqrt{x + 3}} = \frac{4}{1}$

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Solution

$\frac{\sqrt{4 x + 1} + \sqrt{x + 3}}{\sqrt{4 x + 1} - \sqrt{x + 3}} = \frac{4}{1}$
$\Rightarrow \sqrt{4 x + 1} + \sqrt{x + 3} = 4 \sqrt{4 x + 1} - 4 \sqrt{x + 3}$
$\Rightarrow \sqrt{x + 3} + 4 \sqrt{x + 3} = 4 \sqrt{4 x + 1} - \sqrt{4 x + 1}$
$\Rightarrow 5 \sqrt{(x + 3)} = 3 (\sqrt{4 x + 1})$
Squaring both sides
$25 (x + 3) = 9 (4 x + 1)$
$25 x + 75 = 36 x + 9$
$75 - 9 = 36 x - 25 x$
$\Rightarrow 66 = 11 x$
$\Rightarrow x = \frac{66}{11} = 6$

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