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Integration by Substitution

Solve the fol...

Question

Solve the following equation:
$\sqrt{3 x - 5} = 3 - \sqrt{x - 2}$

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Solution

$\sqrt{3 x - 5} = 3 - \sqrt{x - 2}$
squaring on both sides
$3 x - 5 = (3 - \sqrt{x - 2})^{2}$
$3 x - 5 = 9 + x - 2 - 6 \sqrt{x - 2}$
$(2 x - 12) = - 6 \sqrt{x - 2}$
$(x - 6)^{2} = 9 (x - 2)$
$x^{2} - 12 x + 36 = 9 x - 18$
$x^{2} - 21 x + 54 = 0 ⟹ x = 3, 18$
But $x = 18$ does not satisfy $\sqrt{3 x - 5} = 3 - \sqrt{x - 2}$
$⟹ x = 3$

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