Solve the following quadratic equation by factorisation method: $x^{2} - 2 a x + a^{2} - b^{2} = 0$

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Solution

$x^{2} - 2 a x + (a^{2} - b^{2}) = 0 ⟹ (x^{2} - 2 a x + a^{2}) - b^{2} = 0$
$⟹ (x - a)^{2} - b^{2} = 0$ (by applying $a^{2} - 2 a b + b^{2} = (a - b)^{2}$ formula)
this is of the form $a^{2} - b^{2} = (a - b) (a + b)$
Hence $(x - a)^{2} - b^{2} = 0$ becomes
$(x - a - b) (x - a + b) = 0 ⟹ x - a - b = 0, x - a + b = 0 ⟹ x = a + b, x = a - b$
Hence these are the required roots of the given equation.

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