Question

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

Answer 1

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

squaring both sides -

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

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

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

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

$24-4=4x$

$4x=20$

$\Rightarrow x=5.$

Hence, the answer is $5.$