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

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

Solve for x

$$x = \frac{\sqrt{2 {(\sqrt{37} - 1)}}}{2} \approx 1.594171028$$

$$x = -\frac{\sqrt{2 {(\sqrt{37} - 1)}}}{2} \approx -1.594171028$$

Solution Steps

Square both sides of the equation.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}=\left(\sqrt{9-x^{2}}\right)^{2}$$

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

$$x^{4}=9-x^{2}$$

Subtract 9 from both sides.

$$x^{4}-9=-x^{2}$$

Add x^{2} to both sides.

$$x^{4}-9+x^{2}=0$$

Substitute t for x^{2}.

$$t^{2}+t-9=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 1 for b, and -9 for c in the quadratic formula.

$$t=\frac{-1±\sqrt{1^{2}-4\times 1\left(-9\right)}}{2}$$

Do the calculations.

$$t=\frac{-1±\sqrt{37}}{2}$$

Solve the equation t=\frac{-1±\sqrt{37}}{2} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{37}-1}{2} t=\frac{-\sqrt{37}-1}{2}$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for positive t.

$$x=\frac{\sqrt{2\sqrt{37}-2}}{2} x=-\frac{\sqrt{2\sqrt{37}-2}}{2}$$

Substitute \frac{\sqrt{2\sqrt{37}-2}}{2} for x in the equation x^{2}=\sqrt{9-x^{2}}.

$$\left(\frac{\sqrt{2\sqrt{37}-2}}{2}\right)^{2}=\sqrt{9-\left(\frac{\sqrt{2\sqrt{37}-2}}{2}\right)^{2}}$$

Simplify. The value x=\frac{\sqrt{2\sqrt{37}-2}}{2} satisfies the equation.

$$\frac{1}{2}\times 37^{\frac{1}{2}}-\frac{1}{2}=\frac{1}{2}\times 37^{\frac{1}{2}}-\frac{1}{2}$$

Substitute -\frac{\sqrt{2\sqrt{37}-2}}{2} for x in the equation x^{2}=\sqrt{9-x^{2}}.

$$\left(-\frac{\sqrt{2\sqrt{37}-2}}{2}\right)^{2}=\sqrt{9-\left(-\frac{\sqrt{2\sqrt{37}-2}}{2}\right)^{2}}$$

Simplify. The value x=-\frac{\sqrt{2\sqrt{37}-2}}{2} satisfies the equation.

$$\frac{1}{2}\times 37^{\frac{1}{2}}-\frac{1}{2}=\frac{1}{2}\times 37^{\frac{1}{2}}-\frac{1}{2}$$

List all solutions of x^{2}=\sqrt{9-x^{2}}.

$$x=\frac{\sqrt{2\sqrt{37}-2}}{2} x=-\frac{\sqrt{2\sqrt{37}-2}}{2}$$