For what value of $k$ , will the equation $x^{2} - (3 k - 1) x + 2 k^{2} + 2 k = 11$ have equal roots?

5

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9

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Both (a) and (b)

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0

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Solution

The correct option is D Both (a) and (b)
Given equation is $x^{2} - (3 k - 1) x + 2 k^{2} + 2 k - 11 = 0$
For equal roots, $b^{2} - 4 a c = 0$
Therefore, ${(3 k - 1)}^{2} = 4 (2 k^{2} + 2 k - 11)$
$\Rightarrow 9 k^{2} + 1 - 6 k = 8 k^{2} + 8 k - 44$
$\Rightarrow k^{2} - 14 k + 45 = 0$
$\Rightarrow (k - 5) (k - 9) = 0$
$\Rightarrow k = 5$ or $k = 9$

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

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