Question

Solve each of the following equations by using the method of completing the square:

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

Answer 1

$$\begin{gathered} 5x^2-6x-2=0 \\ \Rightarrow 25x^2-30x-10=0\left(\mathrm{Multiplying}\mathrm{both}\mathrm{sides}\mathrm{by}5\right) \\ \Rightarrow 25x^2-30x=10 \\ \Rightarrow {\left(5x\right)}^2-2\times 5x\times 3+3^2=10+3^2\left(\mathrm{Adding}{3}^2\mathrm{on}\mathrm{both}\mathrm{sides}\right) \end{gathered}$$
$$\begin{gathered} \Rightarrow {\left(5x-3\right)}^2=10+9=19 \\ \Rightarrow 5x-3=\pm \sqrt{19}\left(\mathrm{Taking}\mathrm{square}\mathrm{root}\mathrm{on}\mathrm{both}\mathrm{sides}\right) \\ \Rightarrow 5x-3=\sqrt{19}\mathrm{or}5x-3=-\sqrt{19} \\ \Rightarrow 5x=3+\sqrt{19}\mathrm{or}5x=3-\sqrt{19} \end{gathered}$$
$\Rightarrow x=\frac{3+\sqrt{19}}{5}\mathrm{or}x=\frac{3-\sqrt{19}}{5}$
Hence, $\frac{3+\sqrt{19}}{5}$ and $\frac{3-\sqrt{19}}{5}$ are the roots of the given equation.