Solve 3x^4-96x^2-256=0

$$3 x ^ { 4 } - 96 x ^ { 2 } - 256 = 0$$

Solve for x (complex solution)

$$x=\frac{2\times 3^{\frac{3}{4}}\left(\sqrt{6}-\sqrt{2}\right)i}{3}\approx 1.573279573i$$

$$x=-\frac{2\times 3^{\frac{3}{4}}\left(\sqrt{6}-\sqrt{2}\right)i}{3}\approx -0-1.573279573i$$

$$x = -\frac{2 \cdot 3 ^ {\frac{3}{4}} {(\sqrt{2} + \sqrt{6})}}{3} \approx -5.8715593$$

$$x = \frac{2 \cdot 3 ^ {\frac{3}{4}} {(\sqrt{2} + \sqrt{6})}}{3} \approx 5.8715593$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$3t^{2}-96t-256=0$$

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, -96 for b, and -256 for c in the quadratic formula.

$$t=\frac{-\left(-96\right)±\sqrt{\left(-96\right)^{2}-4\times 3\left(-256\right)}}{2\times 3}$$

Do the calculations.

$$t=\frac{96±64\sqrt{3}}{6}$$

Solve the equation t=\frac{96±64\sqrt{3}}{6} when ± is plus and when ± is minus.

$$t=\frac{32\sqrt{3}}{3}+16 t=-\frac{32\sqrt{3}}{3}+16$$

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

$$x=-\sqrt{\frac{32\sqrt{3}}{3}+16} x=\sqrt{\frac{32\sqrt{3}}{3}+16} x=-i\sqrt{-\left(-\frac{32\sqrt{3}}{3}+16\right)} x=i\sqrt{-\left(-\frac{32\sqrt{3}}{3}+16\right)}$$

Solve for x

$$x = -\frac{2 \cdot 3 ^ {\frac{3}{4}} {(\sqrt{2} + \sqrt{6})}}{3} \approx -5.8715593$$

$$x = \frac{2 \cdot 3 ^ {\frac{3}{4}} {(\sqrt{2} + \sqrt{6})}}{3} \approx 5.8715593$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$3t^{2}-96t-256=0$$

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, -96 for b, and -256 for c in the quadratic formula.

$$t=\frac{-\left(-96\right)±\sqrt{\left(-96\right)^{2}-4\times 3\left(-256\right)}}{2\times 3}$$

Do the calculations.

$$t=\frac{96±64\sqrt{3}}{6}$$

Solve the equation t=\frac{96±64\sqrt{3}}{6} when ± is plus and when ± is minus.

$$t=\frac{32\sqrt{3}}{3}+16 t=-\frac{32\sqrt{3}}{3}+16$$

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

$$x=4\sqrt{\frac{2\sqrt{3}}{3}+1} x=-4\sqrt{\frac{2\sqrt{3}}{3}+1}$$