Solve 12x^4+12x^2-1=0

$$12 { x }^{ 4 } +12 { x }^{ 2 } -1 = 0$$

Solve for x (complex solution)

$$x=\frac{3^{\frac{3}{4}}i}{6}+\frac{\sqrt[4]{3}i}{2}\approx 1.037954849i$$

$$x=-\frac{3^{\frac{3}{4}}i}{6}-\frac{\sqrt[4]{3}i}{2}\approx -1.037954849i$$

$$x=-\frac{\sqrt{12\sqrt{3}-18}}{6}\approx -0.278119164$$

$$x=\frac{\sqrt{12\sqrt{3}-18}}{6}\approx 0.278119164$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$12t^{2}+12t-1=0$$

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 12 for a, 12 for b, and -1 for c in the quadratic formula.

$$t=\frac{-12±\sqrt{12^{2}-4\times 12\left(-1\right)}}{2\times 12}$$

Do the calculations.

$$t=\frac{-12±8\sqrt{3}}{24}$$

Solve the equation t=\frac{-12±8\sqrt{3}}{24} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{3}}{3}-\frac{1}{2} t=-\frac{\sqrt{3}}{3}-\frac{1}{2}$$

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

$$x=-\sqrt{\frac{\sqrt{3}}{3}-\frac{1}{2}} x=\sqrt{\frac{\sqrt{3}}{3}-\frac{1}{2}} x=-i\sqrt{\frac{\sqrt{3}}{3}+\frac{1}{2}} x=i\sqrt{\frac{\sqrt{3}}{3}+\frac{1}{2}}$$

Solve for x

$$x=-\frac{\sqrt{12\sqrt{3}-18}}{6}\approx -0.278119164$$

$$x=\frac{\sqrt{12\sqrt{3}-18}}{6}\approx 0.278119164$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$12t^{2}+12t-1=0$$

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 12 for a, 12 for b, and -1 for c in the quadratic formula.

$$t=\frac{-12±\sqrt{12^{2}-4\times 12\left(-1\right)}}{2\times 12}$$

Do the calculations.

$$t=\frac{-12±8\sqrt{3}}{24}$$

Solve the equation t=\frac{-12±8\sqrt{3}}{24} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{3}}{3}-\frac{1}{2} t=-\frac{\sqrt{3}}{3}-\frac{1}{2}$$

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

$$x=\frac{\sqrt{\frac{4\sqrt{3}}{3}-2}}{2} x=-\frac{\sqrt{\frac{4\sqrt{3}}{3}-2}}{2}$$