Question

Find the square root of
$2a-\sqrt{3a^2-2ab-b^2}$

Answer 1

$2a-\sqrt{3a^2-2ab-b^2}$

$=2a-\sqrt{3a^2-3ab+ab-b^2}$

$=2a-\sqrt{(3a+b)(a-b)}$

$=2a-2\sqrt{\frac{(3a+b)}{2}\times \frac{(a-b)}{2}}$

$=\frac{3a+b}{2}+\frac{a-b}{2}-2\sqrt{\frac{(3a+b)}{2}\times \frac{(a-b)}{2}}$

$={(\sqrt{\frac{3a+b}{2}}-\sqrt{\frac{a-b}{2}})}^2$

Hence, square of $2a-\sqrt{3a^2-2ab-b^2}$ is $=\sqrt{\frac{3a+b}{2}}-\sqrt{\frac{a-b}{2}}$.