Solve x^4+1/x^4=119

$${ x }^{ 4 } + \frac{ 1 }{ { x }^{ 4 } } =119$$

Solve for x (complex solution)

$$x\in \frac{-\sqrt{13}-3}{2},-\frac{\sqrt{13}i}{2}-\frac{3}{2}i,\frac{\sqrt{13}i}{2}+\frac{3}{2}i,\frac{\sqrt{13}+3}{2},-\frac{\sqrt{13}i}{2}+\frac{3}{2}i,\frac{3-\sqrt{13}}{2},\frac{\sqrt{13}i}{2}-\frac{3}{2}i,\frac{\sqrt{13}-3}{2}$$

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}.

$$x^{4}x^{4}+1=119x^{4}$$

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

$$x^{8}+1=119x^{4}$$

Subtract 119x^{4} from both sides.

$$x^{8}+1-119x^{4}=0$$

Substitute t for x^{4}.

$$t^{2}-119t+1=0$$

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 1 for a, -119 for b, and 1 for c in the quadratic formula.

$$t=\frac{-\left(-119\right)±\sqrt{\left(-119\right)^{2}-4\times 1\times 1}}{2}$$

Do the calculations.

$$t=\frac{119±33\sqrt{13}}{2}$$

Solve the equation t=\frac{119±33\sqrt{13}}{2} when ± is plus and when ± is minus.

$$t=\frac{33\sqrt{13}+119}{2} t=\frac{119-33\sqrt{13}}{2}$$

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

$$x=-\left(\frac{\sqrt{13}i}{2}+\frac{3}{2}i\right) x=-\frac{\sqrt{13}+3}{2} x=\frac{\sqrt{13}i}{2}+\frac{3}{2}i x=\frac{\sqrt{13}+3}{2} x=-\frac{\sqrt{13}i}{2}+\frac{3}{2}i x=\frac{3-\sqrt{13}}{2} x=-\left(-\frac{\sqrt{13}i}{2}+\frac{3}{2}i\right) x=-\frac{3-\sqrt{13}}{2}$$

Solve for x

$$x=\frac{-\sqrt{13}-3}{2}\approx -3.302775638$$

$$x = \frac{\sqrt{13} + 3}{2} \approx 3.302775638$$

$$x=\frac{\sqrt{13}-3}{2}\approx 0.302775638$$

$$x=\frac{3-\sqrt{13}}{2}\approx -0.302775638$$

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}.

$$x^{4}x^{4}+1=119x^{4}$$

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

$$x^{8}+1=119x^{4}$$

Subtract 119x^{4} from both sides.

$$x^{8}+1-119x^{4}=0$$

Substitute t for x^{4}.

$$t^{2}-119t+1=0$$

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 1 for a, -119 for b, and 1 for c in the quadratic formula.

$$t=\frac{-\left(-119\right)±\sqrt{\left(-119\right)^{2}-4\times 1\times 1}}{2}$$

Do the calculations.

$$t=\frac{119±33\sqrt{13}}{2}$$

Solve the equation t=\frac{119±33\sqrt{13}}{2} when ± is plus and when ± is minus.

$$t=\frac{33\sqrt{13}+119}{2} t=\frac{119-33\sqrt{13}}{2}$$

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

$$x=\frac{\sqrt{13}+3}{2} x=-\frac{\sqrt{13}+3}{2} x=-\frac{3-\sqrt{13}}{2} x=\frac{3-\sqrt{13}}{2}$$