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Expansion of (x^2 - y^2)

The value of ...

Question

The value of x from the equation
$\sqrt{x^{2} + 2 x - 3} + \sqrt{x^{2} - x} = \sqrt{5 (x - 1)}$ will be ____________.

A

{1, \frac{1}{5}}

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B

{\frac{1}{5}, \frac{1}{3}}

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C

{1, - \frac{1}{5}}

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D

{\frac{1}{2}, \frac{1}{5}}

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Solution

The correct option is A ${1, \frac{1}{5}}$
$\sqrt{x^{2} + 2 x - 3} + \sqrt{x^{2} - x} - \sqrt{5 (x - 1)} = 0$
$\sqrt{(x + 3) (x + 1)} + \sqrt{x (x - 1)} - \sqrt{5 (x - 1)} = 0$
$\sqrt{x - 1} [\sqrt{x + 3} + \sqrt{x} - \sqrt{5}] = 0$
$\sqrt{x - 1} = 0 ⟹ x - 1 = 0 ⟹ x = 1$
and when $\sqrt{x + 3} = \sqrt{5} - \sqrt{x}$
$x + 3 = 5 + x - 2 \sqrt{5 x}$
$2 \sqrt{5 x} = 2 ⟹ \sqrt{5 x} = 1 ⟹ 5 x = 1$
$x = \frac{1}{5}$
$∴ x = 1, \frac{1}{5}$

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