Solve x^4+1/x^4=322

$${ x }^{ 4 } + \frac{ 1 }{ { x }^{ 4 } } =322$$

Solve for x (complex solution)

$$x\in -\left(\sqrt{5}+2\right),-\sqrt{5}i-2i,\sqrt{5}i+2i,\sqrt{5}+2,-\sqrt{5}i+2i,2-\sqrt{5},\sqrt{5}i-2i,\sqrt{5}-2$$

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}.

$$x^{4}x^{4}+1=322x^{4}$$

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

$$x^{8}+1=322x^{4}$$

Subtract 322x^{4} from both sides.

$$x^{8}+1-322x^{4}=0$$

Substitute t for x^{4}.

$$t^{2}-322t+1=0$$

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, -322 for b, and 1 for c in the quadratic formula.

$$t=\frac{-\left(-322\right)±\sqrt{\left(-322\right)^{2}-4\times 1\times 1}}{2}$$

Do the calculations.

$$t=\frac{322±144\sqrt{5}}{2}$$

Solve the equation t=\frac{322±144\sqrt{5}}{2} when ± is plus and when ± is minus.

$$t=72\sqrt{5}+161 t=161-72\sqrt{5}$$

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

$$x=-\left(\sqrt{5}i+2i\right) x=-\left(\sqrt{5}+2\right) x=\sqrt{5}i+2i x=\sqrt{5}+2 x=-\sqrt{5}i+2i x=2-\sqrt{5} x=-\left(-\sqrt{5}i+2i\right) x=-\left(2-\sqrt{5}\right)$$

Solve for x

$$x=-\left(\sqrt{5}+2\right)\approx -4.236067977$$

$$x=\sqrt{5}+2\approx 4.236067977$$

$$x=\sqrt{5}-2\approx 0.236067977$$

$$x=2-\sqrt{5}\approx -0.236067977$$

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}.

$$x^{4}x^{4}+1=322x^{4}$$

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

$$x^{8}+1=322x^{4}$$

Subtract 322x^{4} from both sides.

$$x^{8}+1-322x^{4}=0$$

Substitute t for x^{4}.

$$t^{2}-322t+1=0$$

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, -322 for b, and 1 for c in the quadratic formula.

$$t=\frac{-\left(-322\right)±\sqrt{\left(-322\right)^{2}-4\times 1\times 1}}{2}$$

Do the calculations.

$$t=\frac{322±144\sqrt{5}}{2}$$

Solve the equation t=\frac{322±144\sqrt{5}}{2} when ± is plus and when ± is minus.

$$t=72\sqrt{5}+161 t=161-72\sqrt{5}$$

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

$$x=\sqrt{5}+2 x=-\left(\sqrt{5}+2\right) x=-\left(2-\sqrt{5}\right) x=2-\sqrt{5}$$