Question

Solve the quadratic equation $5x^2-6x-2=0$ by completing the square.

Answer 1

Completing the square method:

Given the quadratic equation is:

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

We can write it as:

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

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

Adding ${\left(\frac{3}{5}\right)}^2=\frac{9}{25}$ on both the sides.

$$\begin{gathered} x^2-\frac{6}{5}x+\frac{9}{25}=\frac{2}{5}+\frac{9}{25} \\ \Rightarrow x^2-2\times \left(\frac{3}{5}\right)\times x+{\left(\frac{3}{5}\right)}^2=\frac{10+9}{25} \\ \Rightarrow {(x-\frac{3}{5})}^2=\frac{19}{25} \\ \Rightarrow x-\frac{3}{5}=\pm \sqrt{\frac{19}{25}} \\ \Rightarrow x=\pm \sqrt{\frac{19}{25}}+\frac{3}{5} \\ \Rightarrow x=\frac{3+\sqrt{19}}{5}\text{or}x=\frac{3-\sqrt{19}}{5} \end{gathered}$$

Hence, $\mathit{x}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{3}\mathbf{}\mathbf{+}\mathbf{}\sqrt{\mathbf{19}}}{\mathbf{5}}\mathbf{}\mathbf{\text{or}}\mathit{x}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{3}\mathbf{}\mathbf{-}\mathbf{}\sqrt{\mathbf{19}}}{\mathbf{5}}$ are the solution of given quadratic equation.