Find the roots of the equation $5 x^{2} - 6 x - 2 = 0$ by the method of completing the square.

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\frac{3 \pm \sqrt{19}}{4}

\frac{3 \pm \sqrt{76}}{10}

\frac{3 \pm \sqrt{19}}{5}

\frac{3 \pm \sqrt{76}}{2}

answer is D.

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Detailed Solution

Given equation: $5 x^{2} - 6 x - 2 = 0$

Using the completing square method, we first divide the equation by 5 then try to make the left side of the equation a perfect square.

$5 x^{2} - 6 x - 2 = 0$

$\Rightarrow x^{2} - \frac{6}{5} x - \frac{2}{5} = 0$

$\Rightarrow x^{2} - \frac{6}{5} x = \frac{2}{5}$

$\Rightarrow x^{2} - 2 \times (x) \times (\frac{3}{5}) + {(\frac{3}{5})}^{2} = \frac{2}{5} + {(\frac{3}{5})}^{2}$

$\Rightarrow {(x - \frac{3}{5})}^{2} = \frac{2}{5} + \frac{9}{25}$

$\Rightarrow {(x - \frac{3}{5})}^{2} = \frac{10}{25} + \frac{9}{25}$

$\Rightarrow {(x - \frac{3}{5})}^{2} = \frac{19}{25}$

$\Rightarrow x - \frac{3}{5} = \pm \frac{\sqrt{19}}{5}$

$\Rightarrow x = \frac{3 \pm \sqrt{19}}{5}$

Hence, the value of the roots is $\frac{3 \pm \sqrt{19}}{5}$ .

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