Find the values of $m$ , if the equation $x^{2} - 15 - m (2 x - 8) = 0$ have equal roots

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Solution

Given quadratic equation can be written as,
$x^{2} - (2 m) x + (8 m - 15) = 0$
Now for equal roots, Discriminant $= 0$
$\Rightarrow b^{2} - 4 a c = 0$
$\Rightarrow (- 2 m)^{2} - 4 (8 m - 15) = 0$
$\Rightarrow m^{2} - 8 m + 15 = 0$
$\Rightarrow (m - 3) (m - 5) = 0$ , factor
$\Rightarrow m = 3, 5$

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