Question

Solve the following equation:
$1+\sqrt{1+x\sqrt{x^2-24}}=x\cdot$

Answer 1

$1+\sqrt{1+x\sqrt{x^2+24}}=x$

$\Rightarrow \sqrt{1+x\sqrt{x^2+24}}=x-1$

$1+x\sqrt{x^2+24}={(x-1)}^2$ ( squaring both sides )

$1+x\sqrt{x^2-24}=x^2+1-2x$

$x\sqrt{x^2-24}=x(x-2)$

$x^2-24={(x-2)}^2=x^2+4-4x$

$-24-4=-4x$

$4x=28$

$\Rightarrow x=7.$

Hence, the answer is $7.$