Solve x^4+1/x^4=34

$$x ^ { 4 } + \frac { 1 } { x ^ { 4 } } = 34$$

Solve for x (complex solution)

$$x\in -\sqrt{2}-1,-\sqrt{2}i-i,\sqrt{2}i+i,\sqrt{2}+1,-\sqrt{2}i+i,1-\sqrt{2},\sqrt{2}i-i,\sqrt{2}-1$$

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}.

$$x^{4}x^{4}+1=34x^{4}$$

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

$$x^{8}+1=34x^{4}$$

Subtract 34x^{4} from both sides.

$$x^{8}+1-34x^{4}=0$$

Substitute t for x^{4}.

$$t^{2}-34t+1=0$$

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, -34 for b, and 1 for c in the quadratic formula.

$$t=\frac{-\left(-34\right)±\sqrt{\left(-34\right)^{2}-4\times 1\times 1}}{2}$$

Do the calculations.

$$t=\frac{34±24\sqrt{2}}{2}$$

Solve the equation t=\frac{34±24\sqrt{2}}{2} when ± is plus and when ± is minus.

$$t=12\sqrt{2}+17 t=17-12\sqrt{2}$$

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

$$x=-\left(\sqrt{2}i+i\right) x=-\left(\sqrt{2}+1\right) x=\sqrt{2}i+i x=\sqrt{2}+1 x=-\sqrt{2}i+i x=1-\sqrt{2} x=-\left(-\sqrt{2}i+i\right) x=-\left(1-\sqrt{2}\right)$$

Solve for x

$$x=-\left(\sqrt{2}+1\right)\approx -2.414213562$$

$$x=\sqrt{2}+1\approx 2.414213562$$

$$x=\sqrt{2}-1\approx 0.414213562$$

$$x=1-\sqrt{2}\approx -0.414213562$$

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}.

$$x^{4}x^{4}+1=34x^{4}$$

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

$$x^{8}+1=34x^{4}$$

Subtract 34x^{4} from both sides.

$$x^{8}+1-34x^{4}=0$$

Substitute t for x^{4}.

$$t^{2}-34t+1=0$$

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, -34 for b, and 1 for c in the quadratic formula.

$$t=\frac{-\left(-34\right)±\sqrt{\left(-34\right)^{2}-4\times 1\times 1}}{2}$$

Do the calculations.

$$t=\frac{34±24\sqrt{2}}{2}$$

Solve the equation t=\frac{34±24\sqrt{2}}{2} when ± is plus and when ± is minus.

$$t=12\sqrt{2}+17 t=17-12\sqrt{2}$$

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

$$x=\sqrt{2}+1 x=-\left(\sqrt{2}+1\right) x=-\left(1-\sqrt{2}\right) x=1-\sqrt{2}$$