Solve 0=10/3x^4-1/3x^2-3

$$0 = \frac { 10 } { 3 } x ^ { 4 } - \frac { 1 } { 3 } x ^ { 2 } - 3$$

Solve for x (complex solution)

$$x=1$$

$$x=-1$$

$$x=-\frac{3\sqrt{10}i}{10}\approx -0-0.948683298i$$

$$x=\frac{3\sqrt{10}i}{10}\approx 0.948683298i$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$\frac{10}{3}x^{4}-\frac{1}{3}x^{2}-3=0$$

Substitute t for x^{2}.

$$\frac{10}{3}t^{2}-\frac{1}{3}t-3=0$$

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 \frac{10}{3} for a, -\frac{1}{3} for b, and -3 for c in the quadratic formula.

$$t=\frac{-\left(-\frac{1}{3}\right)±\sqrt{\left(-\frac{1}{3}\right)^{2}-4\times \frac{10}{3}\left(-3\right)}}{2\times \frac{10}{3}}$$

Do the calculations.

$$t=\frac{\frac{1}{3}±\frac{19}{3}}{\frac{20}{3}}$$

Solve the equation t=\frac{\frac{1}{3}±\frac{19}{3}}{\frac{20}{3}} when ± is plus and when ± is minus.

$$t=1 t=-\frac{9}{10}$$

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

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

Solve for x

$$x=-1$$

$$x=1$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$\frac{10}{3}x^{4}-\frac{1}{3}x^{2}-3=0$$

Substitute t for x^{2}.

$$\frac{10}{3}t^{2}-\frac{1}{3}t-3=0$$

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 \frac{10}{3} for a, -\frac{1}{3} for b, and -3 for c in the quadratic formula.

$$t=\frac{-\left(-\frac{1}{3}\right)±\sqrt{\left(-\frac{1}{3}\right)^{2}-4\times \frac{10}{3}\left(-3\right)}}{2\times \frac{10}{3}}$$

Do the calculations.

$$t=\frac{\frac{1}{3}±\frac{19}{3}}{\frac{20}{3}}$$

Solve the equation t=\frac{\frac{1}{3}±\frac{19}{3}}{\frac{20}{3}} when ± is plus and when ± is minus.

$$t=1 t=-\frac{9}{10}$$

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

$$x=1 x=-1$$