Solve x^2-2=8/x^2

$$x ^ { 2 } - 2 = \frac { 8 } { x ^ { 2 } }$$

Solve for x (complex solution)

$$x=\sqrt{2}i\approx 1.414213562i$$

$$x=-\sqrt{2}i\approx -0-1.414213562i$$

$$x=-2$$

$$x=2$$

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^{2}.

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

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

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

Subtract 8 from both sides.

$$x^{4}+x^{2}\left(-2\right)-8=0$$

Substitute t for x^{2}.

$$t^{2}-2t-8=0$$

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 -8 for c in the quadratic formula.

$$t=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 1\left(-8\right)}}{2}$$

Do the calculations.

$$t=\frac{2±6}{2}$$

Solve the equation t=\frac{2±6}{2} when ± is plus and when ± is minus.

$$t=4 t=-2$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for each t.

$$x=-2 x=2 x=-\sqrt{2}i x=\sqrt{2}i$$

Solve for x

$$x=2$$

$$x=-2$$

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^{2}.

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

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

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

Subtract 8 from both sides.

$$x^{4}+x^{2}\left(-2\right)-8=0$$

Substitute t for x^{2}.

$$t^{2}-2t-8=0$$

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 -8 for c in the quadratic formula.

$$t=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 1\left(-8\right)}}{2}$$

Do the calculations.

$$t=\frac{2±6}{2}$$

Solve the equation t=\frac{2±6}{2} when ± is plus and when ± is minus.

$$t=4 t=-2$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for positive t.

$$x=2 x=-2$$