Solve x^2=2sqrt{1-x^2}

$$x ^ { 2 } = 2 \sqrt { 1 - x ^ { 2 } }$$

Solve for x

$$x=\sqrt{2\sqrt{2}-2}\approx 0.910179721$$

$$x=-\sqrt{2\sqrt{2}-2}\approx -0.910179721$$

Solution Steps

Square both sides of the equation.

$$\left(x^{2}\right)^{2}=\left(2\sqrt{1-x^{2}}\right)^{2}$$

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

$$x^{4}=\left(2\sqrt{1-x^{2}}\right)^{2}$$

Expand \left(2\sqrt{1-x^{2}}\right)^{2}.

$$x^{4}=2^{2}\left(\sqrt{1-x^{2}}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$x^{4}=4\left(\sqrt{1-x^{2}}\right)^{2}$$

Calculate \sqrt{1-x^{2}} to the power of 2 and get 1-x^{2}.

$$x^{4}=4\left(1-x^{2}\right)$$

Use the distributive property to multiply 4 by 1-x^{2}.

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

Subtract 4 from both sides.

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

Add 4x^{2} to both sides.

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

Substitute t for x^{2}.

$$t^{2}+4t-4=0$$

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, 4 for b, and -4 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{-4±4\sqrt{2}}{2}$$

Solve the equation t=\frac{-4±4\sqrt{2}}{2} when ± is plus and when ± is minus.

$$t=2\sqrt{2}-2 t=-2\sqrt{2}-2$$

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

$$x=\frac{\sqrt{8\sqrt{2}-8}}{2} x=-\frac{\sqrt{8\sqrt{2}-8}}{2}$$

Substitute \frac{\sqrt{8\sqrt{2}-8}}{2} for x in the equation x^{2}=2\sqrt{1-x^{2}}.

$$\left(\frac{\sqrt{8\sqrt{2}-8}}{2}\right)^{2}=2\sqrt{1-\left(\frac{\sqrt{8\sqrt{2}-8}}{2}\right)^{2}}$$

Simplify. The value x=\frac{\sqrt{8\sqrt{2}-8}}{2} satisfies the equation.

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

Substitute -\frac{\sqrt{8\sqrt{2}-8}}{2} for x in the equation x^{2}=2\sqrt{1-x^{2}}.

$$\left(-\frac{\sqrt{8\sqrt{2}-8}}{2}\right)^{2}=2\sqrt{1-\left(-\frac{\sqrt{8\sqrt{2}-8}}{2}\right)^{2}}$$

Simplify. The value x=-\frac{\sqrt{8\sqrt{2}-8}}{2} satisfies the equation.

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

List all solutions of x^{2}=2\sqrt{1-x^{2}}.

$$x=\frac{\sqrt{8\sqrt{2}-8}}{2} x=-\frac{\sqrt{8\sqrt{2}-8}}{2}$$