Question

Find the square root of
$a+x+\sqrt{2ax+x^2}$

Answer 1

$a+x+\sqrt{2ax+x^2}$

$=a+x+\sqrt{x(2a+x}$

$=a+x+2\sqrt{\frac{x}{2}\times \frac{(2a+x)}{2}}$

$=\frac{x}{2}+\frac{2a+x}{2}+2\sqrt{\frac{x}{2}\times \frac{(2a+x)}{2}}$

$={(\sqrt{\frac{x}{2}}+\sqrt{\frac{2a+x}{2}})}^2$

Hence, square root of $a+x+\sqrt{2ax+x^2}$ is $=\pm (\sqrt{\frac{x}{2}}+\sqrt{\frac{2a+x}{2}})$