Solve {l}{x^4+1/x^4=4-2=2}{x^4+1/x^4=2}{text{levalueof}(x-1/x)}

$$\left. \begin{array} { l } { x ^ { 4 } + \frac { 1 } { x ^ { 4 } } = 4 - 2 = 2 } \\ { x ^ { 4 } + \frac { 1 } { x ^ { 4 } } = 2 } \\ { \text { le value of } ( x - \frac { 1 } { x } ) } \end{array} \right.$$

Solve for x (complex solution)

$$x=i$$

$$x=1$$

$$x=-i$$

$$x=-1$$

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}.

$$x^{4}x^{4}+1=x^{4}\times 4+x^{4}\left(-2\right)$$

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

$$x^{8}+1=x^{4}\times 4+x^{4}\left(-2\right)$$

Combine x^{4}\times 4 and x^{4}\left(-2\right) to get 2x^{4}.

$$x^{8}+1=2x^{4}$$

Subtract 2x^{4} from both sides.

$$x^{8}+1-2x^{4}=0$$

Substitute t for x^{4}.

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

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, -2 for b, and 1 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{2±0}{2}$$

Solutions are the same.

$$t=1$$

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

$$x=1 x=i x=-i x=-1$$

Solve for x

$$x=1$$

$$x=-1$$

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}.

$$x^{4}x^{4}+1=x^{4}\times 4+x^{4}\left(-2\right)$$

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

$$x^{8}+1=x^{4}\times 4+x^{4}\left(-2\right)$$

Combine x^{4}\times 4 and x^{4}\left(-2\right) to get 2x^{4}.

$$x^{8}+1=2x^{4}$$

Subtract 2x^{4} from both sides.

$$x^{8}+1-2x^{4}=0$$

Substitute t for x^{4}.

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

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, -2 for b, and 1 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{2±0}{2}$$

Solutions are the same.

$$t=1$$

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

$$x=-1 x=1$$