Solve the following quadratic equation for $x$ :

$x^{2} - 2 a x - (4 b^{2} - a^{2}) = 0$

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Solution

We have, $x^{2} - 2 a x - (4 b^{2} - a^{2}) = 0$
$\implies$ $x^{2} - 2 a x + a^{2} - 4 b^{2} = 0$

$⟹$ $(x - a)^{2} - (2 b)^{2} = 0$ [Using $a^2-b^2$ = $(a + b) (a - b)$ ]

$∴ (x - a + 2 b) (x - a - 2 b) = 0$

$\Rightarrow x = a - 2 b o r a + 2 b$

Hence, $x = a - 2 b$ or $a + 2 b$

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