Question

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

Answer 1

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

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

Here, $a=1,b=\frac{6}{5},c=\frac{2}{5}$

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

Now, ${\left(\frac{b}{2}\right)}^2={\left(\frac{6}{5\times 2}\right)}^2=\frac{36}{100}$

Adding $\frac{36}{100}$ on both sides,

$\Rightarrow$ $x^2-\frac{6}{5}x+\frac{36}{100}=\frac{2}{5}+\frac{36}{100}$

$\Rightarrow$ ${(x-\frac{6}{10})}^2=\frac{40+36}{100}$

$\Rightarrow$ ${(x-\frac{6}{10})}^2=\frac{76}{100}$

$\Rightarrow$ $(x-\frac{6}{10})=\pm \frac{2\sqrt{19}}{10}$

$\Rightarrow$ $x=\frac{2\sqrt{19}}{10}+\frac{6}{10}$ and $x=-\frac{2\sqrt{19}}{10}+\frac{6}{10}$

$\Rightarrow$ $x=\frac{2(\sqrt{19}+3)}{10}$ and $x=-\frac{2(\sqrt{19}-3)}{10}$

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