Question

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

• A. $\frac{3-\sqrt{19}}{5}$
• B. $\frac{5+\sqrt{19}}{5}$
• C. $\frac{3+\sqrt{19}}{5}$
• D. $\frac{5-\sqrt{19}}{5}$

Answer 1

Answer: (A), (C)

The correct options are
A $\frac{3-\sqrt{19}}{5}$
C $\frac{3+\sqrt{19}}{5}$

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

Dividing throughout by 5, we get

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

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

Adding the square of one-half of the coefficient of x to both sides,

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

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

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

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

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

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

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