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

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

Solve for a

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

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

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

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

Steps Using the Quadratic Formula

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

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

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

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

Subtract 3007a^{4} from both sides.

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

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

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