Solve x=sqrt{9-x^2}+3

$$x= \sqrt{ 9- { x }^{ 2 } } +3$$

Solve for x

$$x=3$$

Solution Steps

Subtract 3 from both sides of the equation.

$$x-3=\sqrt{9-x^{2}}$$

Square both sides of the equation.

$$\left(x-3\right)^{2}=\left(\sqrt{9-x^{2}}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{2}-6x+9=\left(\sqrt{9-x^{2}}\right)^{2}$$

Calculate \sqrt{9-x^{2}} to the power of 2 and get 9-x^{2}.

$$x^{2}-6x+9=9-x^{2}$$

Subtract 9 from both sides.

$$x^{2}-6x+9-9=-x^{2}$$

Subtract 9 from 9 to get 0.

$$x^{2}-6x=-x^{2}$$

Add x^{2} to both sides.

$$x^{2}-6x+x^{2}=0$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}-6x=0$$

Factor out x.

$$x\left(2x-6\right)=0$$

To find equation solutions, solve x=0 and 2x-6=0.

$$x=0 x=3$$

Substitute 0 for x in the equation x=\sqrt{9-x^{2}}+3.

$$0=\sqrt{9-0^{2}}+3$$

Simplify. The value x=0 does not satisfy the equation.

$$0=6$$

Substitute 3 for x in the equation x=\sqrt{9-x^{2}}+3.

$$3=\sqrt{9-3^{2}}+3$$

Simplify. The value x=3 satisfies the equation.

$$3=3$$

Equation x-3=\sqrt{9-x^{2}} has a unique solution.

$$x=3$$