Question

solve the equation
$\sqrt{(x+8)+2\sqrt{(x+7)}}+\sqrt{(x+1)-\sqrt{(x+7)}}=4$
find the value of x.

Answer 1

$\sqrt{(x+8)+2\sqrt{(x+7)}}+\sqrt{(x+1)-\sqrt{(x+7)}}=4$
Put $\sqrt{x+7}=t$
$\Rightarrow \sqrt{t^2+2t+1}+\sqrt{t^2-t-6}=4$
$\Rightarrow |t+1|+\sqrt{(t-3)(t+2)}=4$
$\Rightarrow \sqrt{(t-3)(t+2)}=4-|t+1|$
$\Rightarrow t^2-t-6=14+(t+1{)}^2-8(t+1)$ (taking $t+1\ge 0$)
$\Rightarrow 5t=15$
$\Rightarrow t=3$
$\Rightarrow \sqrt{x+7}=3$
$\Rightarrow x=2$