Solve 81x^4+1/x^4=3007

$$81 x ^ { 4 } + \frac { 1 } { x ^ { 4 } } = 3007$$

Solve for x (complex solution)

$$x\in \frac{-\sqrt{61}-7}{6},-\frac{\sqrt{61}i}{6}-\frac{7}{6}i,\frac{\sqrt{61}i}{6}+\frac{7}{6}i,\frac{\sqrt{61}+7}{6},-\frac{\sqrt{61}i}{6}+\frac{7}{6}i,\frac{7-\sqrt{61}}{6},\frac{\sqrt{61}i}{6}-\frac{7}{6}i,\frac{\sqrt{61}-7}{6}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{4}.

$$81x^{4}x^{4}+1=3007x^{4}$$

To multiply powers of the same base, add their exponents. Add 4 and 4 to get 8.

$$81x^{8}+1=3007x^{4}$$

Subtract 3007x^{4} from both sides.

$$81x^{8}+1-3007x^{4}=0$$

Substitute t for x^{4}.

$$81t^{2}-3007t+1=0$$

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 81 for a, -3007 for b, and 1 for c in the quadratic formula.

$$t=\frac{-\left(-3007\right)±\sqrt{\left(-3007\right)^{2}-4\times 81\times 1}}{2\times 81}$$

Do the calculations.

$$t=\frac{3007±385\sqrt{61}}{162}$$

Solve the equation t=\frac{3007±385\sqrt{61}}{162} when ± is plus and when ± is minus.

$$t=\frac{385\sqrt{61}+3007}{162} t=\frac{3007-385\sqrt{61}}{162}$$

Since x=t^{4}, the solutions are obtained by solving the equation for each t.

$$x=-\left(\frac{\sqrt{61}i}{6}+\frac{7}{6}i\right) x=-\frac{\sqrt{61}+7}{6} x=\frac{\sqrt{61}i}{6}+\frac{7}{6}i x=\frac{\sqrt{61}+7}{6} x=-\frac{\sqrt{61}i}{6}+\frac{7}{6}i x=\frac{7-\sqrt{61}}{6} x=-\left(-\frac{\sqrt{61}i}{6}+\frac{7}{6}i\right) x=-\frac{7-\sqrt{61}}{6}$$

Solve for x

$$x=\frac{-\sqrt{61}-7}{6}\approx -2.468374946$$

$$x = \frac{\sqrt{61} + 7}{6} \approx 2.468374946$$

$$x=\frac{\sqrt{61}-7}{6}\approx 0.135041613$$

$$x=\frac{7-\sqrt{61}}{6}\approx -0.135041613$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{4}.

$$81x^{4}x^{4}+1=3007x^{4}$$

To multiply powers of the same base, add their exponents. Add 4 and 4 to get 8.

$$81x^{8}+1=3007x^{4}$$

Subtract 3007x^{4} from both sides.

$$81x^{8}+1-3007x^{4}=0$$

Substitute t for x^{4}.

$$81t^{2}-3007t+1=0$$

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 81 for a, -3007 for b, and 1 for c in the quadratic formula.

$$t=\frac{-\left(-3007\right)±\sqrt{\left(-3007\right)^{2}-4\times 81\times 1}}{2\times 81}$$

Do the calculations.

$$t=\frac{3007±385\sqrt{61}}{162}$$

Solve the equation t=\frac{3007±385\sqrt{61}}{162} when ± is plus and when ± is minus.

$$t=\frac{385\sqrt{61}+3007}{162} t=\frac{3007-385\sqrt{61}}{162}$$

Since x=t^{4}, the solutions are obtained by evaluating x=±\sqrt[4]{t} for positive t.

$$x=\frac{\sqrt{61}+7}{6} x=-\frac{\sqrt{61}+7}{6} x=-\frac{7-\sqrt{61}}{6} x=\frac{7-\sqrt{61}}{6}$$