Question

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

Answer 1

We have $5x^2-6x-2=0$

$\Rightarrow x^2-\frac{6}{5}x-\frac{2}{5}=0$ (Dividing throughout by $5$)

$\Rightarrow x^2-\frac{6}{5}x=\frac{2}{5}\Rightarrow x^2-2\left(\frac{3}{5}\right)x+{\left(\frac{3}{5}\right)}^2=\frac{2}{5}+{\left(\frac{3}{5}\right)}^2\Rightarrow {(x-\frac{3}{5})}^2=\frac{19}{25}$

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

Hence, the roots of the given equation are $\frac{3+\sqrt{19}}{5}$ and $\frac{3-\sqrt{19}}{5}$