Given the equation
[tex]\sqrt{x+16}=x-\sqrt{x+17}[/tex]
take the square of both sides then move the square root term to one side to get
[tex]\left(\sqrt{x+16}\right)^2=\left(x-\sqrt{x+17}\right)^2[/tex]
[tex]x+16=x^2-2x\sqrt{x+17}+(x+17)[/tex]
[tex]x^2-2x\sqrt{x+17}+1=0[/tex]
[tex]x^2+1=2x\sqrt{x+17}[/tex]
Now take the square of both sides again to get
[tex]\left(x^2+1\right)^2=\left(2x\sqrt{x+17}\right)^2[/tex]
[tex]x^4+2x^2+1=4x^2(x+17)[/tex]
[tex]x^4+2x^2+1=4x^3+68x^2[/tex]
[tex]x^4-4x^3-66x^2+1=0[/tex]
Use a calculator to find 4 solutions,
[tex]x\approx-6.365[/tex]
[tex]x\approx-0.124[/tex]
[tex]x\approx0.123[/tex]
[tex]x\approx10.366[/tex]
Plugging each of these back into the original equation, you would find that the first 3 solutions are extraneous, so the only solution is
[tex]\boxed{x\approx10.366}[/tex]
