Find the values of k for which the given equation has real and equal roots:

$x^{2} - 2 x (1 + 3 k) + 7 (3 + 2 k) = 0$

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Solution

The given equation is $x^{2} - 2 x (1 + 3 k) + 7 (3 + 2 k) = 0$

Here, $a = 1, b = - 2 (1 + 3 k) a n d c = 7 (3 + 2 k)$

$∴ D = b^{2} - 4 a c$

$\Rightarrow D = 4 (3 k + 1)^{2} - 4 \times 1 \times 7 (3 + 2 k)$

$\Rightarrow D = 4 (9 k^{2} + 6 k + 1 - 21 - 14 k)$

$\Rightarrow D = 4 (9 k^{2} - 8 k - 20)$

The given equation will have equal roots, if

$D = 0$

$\Rightarrow 4 (9 k^{2} - 8 k - 20) = 0$

$\Rightarrow 9 k^{2} - 8 k - 20 = 0$

$\Rightarrow 9 k^{2} - 18 k + 10 k - 20 = 0$

$\Rightarrow (k - 2) (9 k + 10) = 0$

$\Rightarrow k - 2 = 0 o r, 9 k + 10 = 0 \Rightarrow k = 2 o r, k = - \frac{10}{9}$

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