Question

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

Answer 1

Given quadratic equation, $5x^2-6x-2=0$

Adjusting first term as square we get : $(\sqrt{5}{)}^2$

Adjusting second terms as $-2ab$ we get : $-2\left(\sqrt{5}x\right)\left(\frac{3}{\sqrt{5}}\right)$

Hence adding and subtracting ${\left(\frac{3}{\sqrt{5}}\right)}^2$ to make the equation of the form perfect square.

Then the equation becomes :

$(\sqrt{5}x{)}^2-2(\sqrt{5}x)\left(\frac{3}{\sqrt{5}}\right)+{\left(\frac{3}{\sqrt{5}}\right)}^2-2-\frac{9}{5}=0$

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

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

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

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