Question

Solve the equation 5$x^2$– 6$x$– 2 = 0 by the method of completing the square. Find the positive root.

Answer 1

Multiplying the equation throughout by 5, we get

25 $x^2$– 30 $x$ – 10= 0

This is the same as

$(5x{)}^2-2\times 5x\times 3+3^2-3^2-10=0$

$\Rightarrow$ ${(5x-3)}^2$ - 9 - 10 = 0

$\Rightarrow$ ${(5x-3)}^2$ - 19 = 0

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

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

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

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

Therefore, the roots are $\frac{3+\sqrt{19}}{5}$ and $\frac{3-\sqrt{19}}{5}$.

Since $\sqrt{19}>3$, $\frac{3-\sqrt{19}}{5}$ will be negative.

Thus, the positive root is $\frac{3+\sqrt{19}}{5}$.