Solve x^4+x^2+x^2=1

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

Solve for x (complex solution)

$$x=i\sqrt{\sqrt{2}+1}\approx 1.553773974i$$

$$x=-i\sqrt{\sqrt{2}+1}\approx -0-1.553773974i$$

$$x=-\sqrt{\sqrt{2}-1}\approx -0.643594253$$

$$x=\sqrt{\sqrt{2}-1}\approx 0.643594253$$

Steps Using the Quadratic Formula

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

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

Subtract 1 from both sides.

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

Substitute t for x^{2}.

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

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, 2 for b, and -1 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{-2±2\sqrt{2}}{2}$$

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

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

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

$$x=-\sqrt{\sqrt{2}-1} x=\sqrt{\sqrt{2}-1} x=-i\sqrt{\sqrt{2}+1} x=i\sqrt{\sqrt{2}+1}$$

Solve for x

$$x=-\sqrt{\sqrt{2}-1}\approx -0.643594253$$

$$x=\sqrt{\sqrt{2}-1}\approx 0.643594253$$

Steps Using the Quadratic Formula

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

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

Subtract 1 from both sides.

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

Substitute t for x^{2}.

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

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, 2 for b, and -1 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{-2±2\sqrt{2}}{2}$$

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

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

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

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