Solve $x^{2} - 2 a x + (a^{2} - b^{2}) = 0$ .

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Solution

Given $x^{2} - 2 a x + a^{2} - b^{2} = 0$
$\Rightarrow x^{2} - a x - a x + a^{2} - b^{2} = 0 \Rightarrow x (x - a) - a (x - a) - b^{2} = 0 \Rightarrow (x - a) (x - a) - b^{2} = 0 \Rightarrow {(x - a)}^{2} - b^{2} = 0 \Rightarrow (x - a - b) (x - a + b) = 0$
Either $x - a - b = 0 or x - a + b = 0$
$\Rightarrow x = a + b or x = a - b$
Hence, $x = a + b or x = a - b$ is the solution of the given quadratic equation.

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