Solve x^2+9/x^2=22

$${ x }^{ 2 } + \frac{ 9 }{ { x }^{ 2 } } =22$$

Solve for x

$$x=-\left(\sqrt{7}+2\right)\approx -4.645751311$$

$$x=\sqrt{7}+2\approx 4.645751311$$

$$x=\sqrt{7}-2\approx 0.645751311$$

$$x=2-\sqrt{7}\approx -0.645751311$$

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}.

$$x^{2}x^{2}+9=22x^{2}$$

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

$$x^{4}+9=22x^{2}$$

Subtract 22x^{2} from both sides.

$$x^{4}+9-22x^{2}=0$$

Substitute t for x^{2}.

$$t^{2}-22t+9=0$$

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, -22 for b, and 9 for c in the quadratic formula.

$$t=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 1\times 9}}{2}$$

Do the calculations.

$$t=\frac{22±8\sqrt{7}}{2}$$

Solve the equation t=\frac{22±8\sqrt{7}}{2} when ± is plus and when ± is minus.

$$t=4\sqrt{7}+11 t=11-4\sqrt{7}$$

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

$$x=\sqrt{7}+2 x=-\left(\sqrt{7}+2\right) x=-\left(2-\sqrt{7}\right) x=2-\sqrt{7}$$