For what value of $k$ does the quadratic equation $(k - 5) x^{2} + 2 (k - 5) x + 2 = 0$ have equal roots?

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Solution

Quadratic equation $a x^{2} + b x + c = 0$ has equal roots if $b^{2} - 4 a c = 0$
$\Rightarrow [2 (k - 5)]^{2} - 4 (2) (k - 5) = 0$
$\Rightarrow 4 (k - 5) [k - 5 - 2] = 0$
$\Rightarrow k = 5$ or $k = 7$

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