Find the values of k so that the quadratic equation $(4 - k) x^{2} + 2 (k + 2) x + (8 k + 1) = 0$ has equal roots.

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Solution

$(4 - k) x^{2} + 2 (k + 2) x + (8 k + 1) = 0$ for equal roots, $D = 0$
$a = (4 - k), b = 2 k + 4, c = 8 k + 1$ , $D = b^{2} - 4 a c$
$D = (2 k + 4)^{2} - 4 (8 k + 1) (4 - k)$
$O = 4 k^{2} + 16 + 16 k - 4 (32 k - 8 k^{2} + 4 - k)$
$O = 4 k^{2} + 16 k + 16 - 128 k + 32 k^{2} - 16 + 4 k$
$O = 36 k^{2} - 108 k$
$O = k (36 k - 108)$
$∴ k = 0, k = 3$
$36 k = 108$
$k = 3$ .

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