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Variable Separable Method

Solve the fol...

Question

Solve the following equation:
$\sqrt{3 x + 3} + \sqrt{3 x - 2} = 7$

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Solution

$\sqrt{3 x + 3} + \sqrt{3 x - 2} = 7$
squaring both sides -
$\Rightarrow 3 x + 3 + 3 x - 2 + 2 \sqrt{3 x + 3} \sqrt{3 x - 2} = 49$
$\Rightarrow 6 x + 1 + 2 \sqrt{3 x + 3} \sqrt{3 x - 2} = 49$
$\Rightarrow 2 \sqrt{3 x + 3} \sqrt{3 x - 2} = 48 - 6 x = 36 (8 - x)$
squaring both sides -
$\Rightarrow (3 x + 3) (3 x - 2) = 9 (64 + x^{2} - 16 x)$
$\Rightarrow 9 x^{2} + 9 x - 6 x - 6 = 9 x^{2} - 144 x + 576$
$\Rightarrow 3 x + 144 x = 576 + 6$
$\Rightarrow 147 x = 582$
$\Rightarrow x = \frac{194}{49}$
Hence, the answer is $\frac{194}{49} .$

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