For what value of k are the roots of the quadratic equation $x^{2} - (3 k - 1) x - 2 k^{2} + 2 k + 11 = 0$ equal.

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Solution

Given,

$x^{2} - (3 k - 1) x - 2 k^{2} + 2 k + 11 = 0$

condition for roots to be equal is,

$b^{2} = 4 a c$

$[- (3 k - 1)]^{2} = 4 (1) (- 2 k^{2} + 2 k + 11)$

$9 k^{2} + 1 - 6 k = - 8 k^{2} + 8 k + 44$

$17 k^{2} - 14 k - 43 = 0$

$k = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

$= \frac{14 \pm \sqrt{(- 14)^{2} - 4 (17) (- 43)}}{2 (17)}$

$∴ k = \frac{7 \pm 2 \sqrt{195}}{17}$

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