Solve x^4+4/x^4=4

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

Solve for x (complex solution)

$$x=\sqrt[4]{2}i\approx 1.189207115i$$

$$x=\sqrt[4]{2}\approx 1.189207115$$

$$x=-\sqrt[4]{2}\approx -1.189207115$$

$$x=-\sqrt[4]{2}i\approx -0-1.189207115i$$

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}.

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

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

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

Subtract 4x^{4} from both sides.

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

Substitute t for x^{4}.

$$t^{2}-4t+4=0$$

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, -4 for b, and 4 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{4±0}{2}$$

Solutions are the same.

$$t=2$$

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

$$x=\sqrt[4]{2} x=\sqrt[4]{2}i x=-\sqrt[4]{2} x=-\sqrt[4]{2}i$$

Solve for x

$$x=\sqrt[4]{2}\approx 1.189207115$$

$$x=-\sqrt[4]{2}\approx -1.189207115$$

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}.

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

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

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

Subtract 4x^{4} from both sides.

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

Substitute t for x^{4}.

$$t^{2}-4t+4=0$$

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, -4 for b, and 4 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{4±0}{2}$$

Solutions are the same.

$$t=2$$

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

$$x=-\sqrt[4]{2} x=\sqrt[4]{2}$$