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Nature of Roots

For what valu...

Question

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

9, - 5

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- 9, 5

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9, 5

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- 9, - 5

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Solution

The correct option is B $9, 5$
For this equation to have equal roots
$b^{2} = 4 a c$ Here $a = 1, b = - (3 k - 1)$ and $c = 2 k^{2} + 2 k - 11$
Then ${(- (3 k - 1))}^{2} = 4 (1 (2 k^{2} + 2 k - 11))$
$\Rightarrow 9 k^{2} - 8 k + 1 = 8 k^{2} + 8 k - 44$
$\Rightarrow 9 k^{2} - 8 k^{2} - 6 k - 8 k + 1 + 44 = 0$
$\Rightarrow k^{2} - 14 k + 45 = 0$
$\Rightarrow k^{2} - 9 k - 5 k + 45 = 0$
$\Rightarrow k (k - 9) - 5 (k - 9) = 0$
$\Rightarrow (k - 9) (k - 5) = 0$
Then $k - 9 = 0 o r k = 9$
Or $k - 5 = 0 o r k = 5$
Then values are $9, 5$

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