Solve x^4-2x^2-1=0

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

Solve for x (complex solution)

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

$$x=-i\sqrt{\sqrt{2}-1}\approx -0-0.643594253i$$

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

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

Steps Using the Quadratic Formula

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.

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

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.

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

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

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

Solve for x

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

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

Steps Using the Quadratic Formula

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.

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

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.

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

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

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