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\sqrt{x+16}=x-4
Subtract 4 from both sides of the equation.
\left(\sqrt{x+16}\right)^{2}=\left(x-4\right)^{2}
Square both sides of the equation.
x+16=\left(x-4\right)^{2}
Calculate \sqrt{x+16} to the power of 2 and get x+16.
x+16=x^{2}-8x+16
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.
x+16-x^{2}=-8x+16
Subtract x^{2} from both sides.
x+16-x^{2}+8x=16
Add 8x to both sides.
9x+16-x^{2}=16
Combine x and 8x to get 9x.
9x+16-x^{2}-16=0
Subtract 16 from both sides.
9x-x^{2}=0
Subtract 16 from 16 to get 0.
x\left(9-x\right)=0
Factor out x.
x=0 x=9
To find equation solutions, solve x=0 and 9-x=0.
\sqrt{0+16}+4=0
Substitute 0 for x in the equation \sqrt{x+16}+4=x.
8=0
Simplify. The value x=0 does not satisfy the equation.
\sqrt{9+16}+4=9
Substitute 9 for x in the equation \sqrt{x+16}+4=x.
9=9
Simplify. The value x=9 satisfies the equation.
x=9
Equation \sqrt{x+16}=x-4 has a unique solution.