Solve 3x^4+2x^2-3=0

$$3 x ^ { 4 } + 2 x ^ { 2 } - 3 = 0$$

Solve for x (complex solution)

$$x=\frac{i\sqrt{3\left(\sqrt{10}+1\right)}}{3}\approx 1.177890439i$$

$$x=-\frac{i\sqrt{3\left(\sqrt{10}+1\right)}}{3}\approx -0-1.177890439i$$

$$x=-\frac{\sqrt{3\left(\sqrt{10}-1\right)}}{3}\approx -0.848975394$$

$$x=\frac{\sqrt{3\left(\sqrt{10}-1\right)}}{3}\approx 0.848975394$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$3t^{2}+2t-3=0$$

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 3 for a, 2 for b, and -3 for c in the quadratic formula.

$$t=\frac{-2±\sqrt{2^{2}-4\times 3\left(-3\right)}}{2\times 3}$$

Do the calculations.

$$t=\frac{-2±2\sqrt{10}}{6}$$

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

$$t=\frac{\sqrt{10}-1}{3} t=\frac{-\sqrt{10}-1}{3}$$

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

$$x=-\sqrt{\frac{\sqrt{10}-1}{3}} x=\sqrt{\frac{\sqrt{10}-1}{3}} x=-i\sqrt{\frac{\sqrt{10}+1}{3}} x=i\sqrt{\frac{\sqrt{10}+1}{3}}$$

Solve for x

$$x=-\frac{\sqrt{3\left(\sqrt{10}-1\right)}}{3}\approx -0.848975394$$

$$x=\frac{\sqrt{3\left(\sqrt{10}-1\right)}}{3}\approx 0.848975394$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$3t^{2}+2t-3=0$$

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 3 for a, 2 for b, and -3 for c in the quadratic formula.

$$t=\frac{-2±\sqrt{2^{2}-4\times 3\left(-3\right)}}{2\times 3}$$

Do the calculations.

$$t=\frac{-2±2\sqrt{10}}{6}$$

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

$$t=\frac{\sqrt{10}-1}{3} t=\frac{-\sqrt{10}-1}{3}$$

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

$$x=\sqrt{\frac{\sqrt{10}-1}{3}} x=-\sqrt{\frac{\sqrt{10}-1}{3}}$$