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\left(\sqrt{x+9}\right)^{2}=\left(\sqrt{x}+1\right)^{2}
Square both sides of the equation.
x+9=\left(\sqrt{x}+1\right)^{2}
Calculate \sqrt{x+9} to the power of 2 and get x+9.
x+9=\left(\sqrt{x}\right)^{2}+2\sqrt{x}+1
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{x}+1\right)^{2}.
x+9=x+2\sqrt{x}+1
Calculate \sqrt{x} to the power of 2 and get x.
x+9-x=2\sqrt{x}+1
Subtract x from both sides.
9=2\sqrt{x}+1
Combine x and -x to get 0.
2\sqrt{x}+1=9
Swap sides so that all variable terms are on the left hand side.
2\sqrt{x}=9-1
Subtract 1 from both sides.
2\sqrt{x}=8
Subtract 1 from 9 to get 8.
\sqrt{x}=\frac{8}{2}
Divide both sides by 2.
\sqrt{x}=4
Divide 8 by 2 to get 4.
x=16
Square both sides of the equation.
\sqrt{16+9}=\sqrt{16}+1
Substitute 16 for x in the equation \sqrt{x+9}=\sqrt{x}+1.
5=5
Simplify. The value x=16 satisfies the equation.
x=16
Equation \sqrt{x+9}=\sqrt{x}+1 has a unique solution.