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

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Solution

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

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

$x^{2} - \frac{6}{5} x + \frac{9}{25} - \frac{9}{25} - \frac{2}{5} = 0$

${(x - \frac{3}{5})}^{2} - [\frac{9}{25} + \frac{2}{5}] = 0$

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

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

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

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

Required roots $= \frac{3 + \sqrt{19}}{5}, \frac{3 - \sqrt{19}}{5}$ .

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