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

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

Solve for x (complex solution)

$$x=\sqrt{3}-1\approx 0.732050808$$

$$x=1-\sqrt{3}\approx -0.732050808$$

$$x=-\left(\sqrt{3}+1\right)\approx -2.732050808$$

$$x=\sqrt{3}+1\approx 2.732050808$$

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.

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

Do the calculations.

$$t=\frac{8±4\sqrt{3}}{2}$$

Solve the equation t=\frac{8±4\sqrt{3}}{2} when ± is plus and when ± is minus.

$$t=2\sqrt{3}+4 t=4-2\sqrt{3}$$

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

$$x=-\left(\sqrt{3}+1\right) x=\sqrt{3}+1 x=1-\sqrt{3} x=-\left(1-\sqrt{3}\right)$$

Solve for x

$$x=1-\sqrt{3}\approx -0.732050808$$

$$x=\sqrt{3}-1\approx 0.732050808$$

$$x=\sqrt{3}+1\approx 2.732050808$$

$$x=-\sqrt{3}-1\approx -2.732050808$$

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.

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

Do the calculations.

$$t=\frac{8±4\sqrt{3}}{2}$$

Solve the equation t=\frac{8±4\sqrt{3}}{2} when ± is plus and when ± is minus.

$$t=2\sqrt{3}+4 t=4-2\sqrt{3}$$

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

$$x=\sqrt{3}+1 x=-\left(\sqrt{3}+1\right) x=-\left(1-\sqrt{3}\right) x=1-\sqrt{3}$$