If the equation $x^{2} - 15 - m (2 x - 8) = 0$ has equal roots, find the values of $m$ .

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Solution

Given that the equation $x^{2} - 15 - m (2 x - 8) = 0$ has equal roots
The given equation can also be written as $x^{2} - 2 m x + 8 m - 15 = 0$
For the equation to have equal roots, the discriminant must be equal to zero
$\Rightarrow b^{2} - 4 a c = (- 2 m)^{2} - 4 (8 m - 15) = 0$
$\Rightarrow 4 m^{2} - 32 m + 60 = 0$
$\Rightarrow m^{2} - 8 m + 15 = 0$
$\Rightarrow m = 3, 5$

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