Solve 5x^4+2x^2-1=0

$$5 x ^ { 4 } + 2 x ^ { 2 } - 1 = 0$$

Solve for x (complex solution)

$$x=\frac{i\sqrt{5\left(\sqrt{6}+1\right)}}{5}\approx 0.830600956i$$

$$x=-\frac{i\sqrt{5\left(\sqrt{6}+1\right)}}{5}\approx -0-0.830600956i$$

$$x=-\frac{\sqrt{5\left(\sqrt{6}-1\right)}}{5}\approx -0.53842172$$

$$x=\frac{\sqrt{5\left(\sqrt{6}-1\right)}}{5}\approx 0.53842172$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$5t^{2}+2t-1=0$$

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

$$t=\frac{-2±\sqrt{2^{2}-4\times 5\left(-1\right)}}{2\times 5}$$

Do the calculations.

$$t=\frac{-2±2\sqrt{6}}{10}$$

Solve the equation t=\frac{-2±2\sqrt{6}}{10} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{6}-1}{5} t=\frac{-\sqrt{6}-1}{5}$$

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

$$x=-\sqrt{\frac{\sqrt{6}-1}{5}} x=\sqrt{\frac{\sqrt{6}-1}{5}} x=-i\sqrt{\frac{\sqrt{6}+1}{5}} x=i\sqrt{\frac{\sqrt{6}+1}{5}}$$

Solve for x

$$x=-\frac{\sqrt{5\left(\sqrt{6}-1\right)}}{5}\approx -0.53842172$$

$$x=\frac{\sqrt{5\left(\sqrt{6}-1\right)}}{5}\approx 0.53842172$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$5t^{2}+2t-1=0$$

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

$$t=\frac{-2±\sqrt{2^{2}-4\times 5\left(-1\right)}}{2\times 5}$$

Do the calculations.

$$t=\frac{-2±2\sqrt{6}}{10}$$

Solve the equation t=\frac{-2±2\sqrt{6}}{10} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{6}-1}{5} t=\frac{-\sqrt{6}-1}{5}$$

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

$$x=\sqrt{\frac{\sqrt{6}-1}{5}} x=-\sqrt{\frac{\sqrt{6}-1}{5}}$$