Solve x^4-8x^2-4=0

$${ x }^{ 4 } -8 { x }^{ 2 } -4=0$$

Solve for x (complex solution)

$$x=i\sqrt{2\left(\sqrt{5}-2\right)}\approx 0.687121499i$$

$$x=-i\sqrt{2\left(\sqrt{5}-2\right)}\approx -0-0.687121499i$$

$$x=-\sqrt{2\left(\sqrt{5}+2\right)}\approx -2.91069338$$

$$x=\sqrt{2\left(\sqrt{5}+2\right)}\approx 2.91069338$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-8t-4=0$$

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, -8 for b, and -4 for c in the quadratic formula.

$$t=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 1\left(-4\right)}}{2}$$

Do the calculations.

$$t=\frac{8±4\sqrt{5}}{2}$$

Solve the equation t=\frac{8±4\sqrt{5}}{2} when ± is plus and when ± is minus.

$$t=2\sqrt{5}+4 t=4-2\sqrt{5}$$

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

$$x=-\sqrt{2\sqrt{5}+4} x=\sqrt{2\sqrt{5}+4} x=-i\sqrt{-\left(4-2\sqrt{5}\right)} x=i\sqrt{-\left(4-2\sqrt{5}\right)}$$

Solve for x

$$x=-\sqrt{2\left(\sqrt{5}+2\right)}\approx -2.91069338$$

$$x=\sqrt{2\left(\sqrt{5}+2\right)}\approx 2.91069338$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-8t-4=0$$

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, -8 for b, and -4 for c in the quadratic formula.

$$t=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 1\left(-4\right)}}{2}$$

Do the calculations.

$$t=\frac{8±4\sqrt{5}}{2}$$

Solve the equation t=\frac{8±4\sqrt{5}}{2} when ± is plus and when ± is minus.

$$t=2\sqrt{5}+4 t=4-2\sqrt{5}$$

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

$$x=\frac{\sqrt{8\sqrt{5}+16}}{2} x=-\frac{\sqrt{8\sqrt{5}+16}}{2}$$