Solve left(x^2+1right)^2-x^2=0

$${ \left( { x }^{ 2 } +1 \right) }^{ 2 } - { x }^{ 2 } =0$$

Solve for x (complex solution)

$$x=e^{\frac{2\pi i}{3}}\approx -0.5+0.866025404i$$

$$x=e^{\frac{5\pi i}{3}}\approx 0.5-0.866025404i$$

$$x=e^{\frac{4\pi i}{3}}\approx -0.5-0.866025404i$$

$$x=e^{\frac{\pi i}{3}}\approx 0.5+0.866025404i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{2}+1\right)^{2}.

$$\left(x^{2}\right)^{2}+2x^{2}+1-x^{2}=0$$

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

$$x^{4}+2x^{2}+1-x^{2}=0$$

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

$$x^{4}+x^{2}+1=0$$

Substitute t for x^{2}.

$$t^{2}+t+1=0$$

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 1 for c in the quadratic formula.

$$t=\frac{-1±\sqrt{1^{2}-4\times 1\times 1}}{2}$$

Do the calculations.

$$t=\frac{-1±\sqrt{-3}}{2}$$

Solve the equation t=\frac{-1±\sqrt{-3}}{2} when ± is plus and when ± is minus.

$$t=\frac{-1+\sqrt{3}i}{2} t=\frac{-\sqrt{3}i-1}{2}$$

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

$$x=-e^{\frac{\pi i}{3}} x=e^{\frac{\pi i}{3}} x=-ie^{\frac{\pi i}{6}} x=ie^{\frac{\pi i}{6}}$$