Solve 3x^4-5x^2-4=0

$$3 x ^ { 4 } - 5 x ^ { 2 } - 4 = 0$$

Solve for x (complex solution)

$$x=\frac{i\sqrt{6\left(\sqrt{73}-5\right)}}{6}\approx 0.768548821i$$

$$x=-\frac{i\sqrt{6\left(\sqrt{73}-5\right)}}{6}\approx -0-0.768548821i$$

$$x = -\frac{\sqrt{6 {(\sqrt{73} + 5)}}}{6} \approx -1.502442664$$

$$x = \frac{\sqrt{6 {(\sqrt{73} + 5)}}}{6} \approx 1.502442664$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$3t^{2}-5t-4=0$$

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 3 for a, -5 for b, and -4 for c in the quadratic formula.

$$t=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 3\left(-4\right)}}{2\times 3}$$

Do the calculations.

$$t=\frac{5±\sqrt{73}}{6}$$

Solve the equation t=\frac{5±\sqrt{73}}{6} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{73}+5}{6} t=\frac{5-\sqrt{73}}{6}$$

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

$$x=-\sqrt{\frac{\sqrt{73}+5}{6}} x=\sqrt{\frac{\sqrt{73}+5}{6}} x=-i\sqrt{-\frac{5-\sqrt{73}}{6}} x=i\sqrt{-\frac{5-\sqrt{73}}{6}}$$

Solve for x

$$x = -\frac{\sqrt{6 {(\sqrt{73} + 5)}}}{6} \approx -1.502442664$$

$$x = \frac{\sqrt{6 {(\sqrt{73} + 5)}}}{6} \approx 1.502442664$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$3t^{2}-5t-4=0$$

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 3 for a, -5 for b, and -4 for c in the quadratic formula.

$$t=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 3\left(-4\right)}}{2\times 3}$$

Do the calculations.

$$t=\frac{5±\sqrt{73}}{6}$$

Solve the equation t=\frac{5±\sqrt{73}}{6} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{73}+5}{6} t=\frac{5-\sqrt{73}}{6}$$

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

$$x=\frac{\sqrt{\frac{2\sqrt{73}+10}{3}}}{2} x=-\frac{\sqrt{\frac{2\sqrt{73}+10}{3}}}{2}$$