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

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\frac{3 \pm \sqrt{21}}{7}

\frac{7 \pm \sqrt{17}}{5}

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

None

answer is C.

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Detailed Solution

Given equation,

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

Divide both side of equation by 5,

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

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

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

\Rightarrow x^{2} - 2 \times x \frac{3}{5} - \frac{2}{5} = 0

After adding and subtracting

{(\frac{3}{5})}^{2}

from left hand side, we get

\Rightarrow x^{2} - 2 \times x \times \frac{3}{5} - \frac{2}{5} + {(\frac{3}{5})}^{2} - {(\frac{3}{5})}^{2} = 0

\Rightarrow x^{2} - 2 \times x \times \frac{3}{5} + {(\frac{3}{5})}^{2} - \frac{2}{5} - {(\frac{3}{5})}^{2} = 0

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

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

\Rightarrow x - \frac{3}{5} = \pm \sqrt{\frac{19}{25}}

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

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

Thus, the roots are,

x = \frac{3 + \sqrt{19}}{5}

and

x = \frac{3 - \sqrt{19}}{5}

Option 3 is Correct.

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