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Integration of Trigonometric Functions

Solve the fol...

Question

Solve the following equation:
$\sqrt{x^{2} - 4 x + 3} + \sqrt{- x^{2} + 3 x - 2} = \sqrt{x^{2} - x}$

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Solution

Squaring on both sides
$x^{2} - 4 x + 3 - x^{2} + 3 x - 2 + 2 \sqrt{(x^{2} - 4 x + 3) (- x^{2} + 3 x - 2)} = x^{2} - x$
$⟹ 2 \sqrt{(x^{2} - 4 x + 3) (- x^{2} + 3 x - 2)} = x^{2} - 1$
Squaring on both sides
$⟹ - x^{4} + 7 x^{3} - 17 x^{2} + 17 x - 6 = x^{4} - 2 x^{2} + 1$
$⟹ 2 x^{4} - 7 x^{3} + 15 x^{2} - 17 x + 7 = 0$
$⟹ x = 1$ is only the real soluiton

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