Question

Roots of the quadratic equation $3x^2-2\sqrt{15}x-2=0$ are

$\frac{\sqrt{15}\pm \sqrt{21}}{3}$. If true answer is 1,else 0

Answer 1

We have $3x^2-2\sqrt{15}x-2=0$
$\Rightarrow x^2-2\frac{\sqrt{15}}{3}x-\frac{2}{3}=0$
[Dividing both sides by the coefficient of $x^2$]
$\Rightarrow x^2-2\frac{\sqrt{15}}{3}x+{\left(\frac{\sqrt{15}}{3}\right)}^2-{\left(\frac{\sqrt{15}}{3}\right)}^2-\frac{2}{3}=0$
[Adding and subtracting the square of half the coefficient of x]
$\Rightarrow {(x-\frac{\sqrt{15}}{3})}^2-\frac{15}{9}+\frac{2}{3}=0$
$\Rightarrow {(x-\frac{\sqrt{15}}{3})}^2-\left(\frac{15+6}{9}\right)=0$
$\Rightarrow {(x-\frac{\sqrt{15}}{3})}^2-\frac{21}{9}=0$
$\Rightarrow x-\frac{\sqrt{15}}{3}=\pm \frac{\sqrt{21}}{3}$
[Taking the square roots on both sides]
$\Rightarrow x=\frac{\sqrt{15}}{3}\pm \frac{\sqrt{21}}{3}=\frac{\sqrt{15}\pm \sqrt{21}}{3}$
Hence the requires solutions are
$x=\frac{\sqrt{15}+\sqrt{21}}{3}andx=\frac{\sqrt{15}-\sqrt{21}}{3}$