Solve 1/x^2+2=x^2-7

$$\frac{ 1 }{ { x }^{ 2 } +2 } = { x }^{ 2 } -7$$

Solve for x

$$x = -\frac{\sqrt{2 {(\sqrt{85} + 5)}}}{2} \approx -2.666415614$$

$$x = \frac{\sqrt{2 {(\sqrt{85} + 5)}}}{2} \approx 2.666415614$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by x^{2}+2.

$$1=\left(x^{2}+2\right)x^{2}+\left(x^{2}+2\right)\left(-7\right)$$

Use the distributive property to multiply x^{2}+2 by x^{2}.

$$1=x^{4}+2x^{2}+\left(x^{2}+2\right)\left(-7\right)$$

Use the distributive property to multiply x^{2}+2 by -7.

$$1=x^{4}+2x^{2}-7x^{2}-14$$

Combine 2x^{2} and -7x^{2} to get -5x^{2}.

$$1=x^{4}-5x^{2}-14$$

Swap sides so that all variable terms are on the left hand side.

$$x^{4}-5x^{2}-14=1$$

Subtract 1 from both sides.

$$x^{4}-5x^{2}-14-1=0$$

Subtract 1 from -14 to get -15.

$$x^{4}-5x^{2}-15=0$$

Substitute t for x^{2}.

$$t^{2}-5t-15=0$$

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, -5 for b, and -15 for c in the quadratic formula.

$$t=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 1\left(-15\right)}}{2}$$

Do the calculations.

$$t=\frac{5±\sqrt{85}}{2}$$

Solve the equation t=\frac{5±\sqrt{85}}{2} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{85}+5}{2} t=\frac{5-\sqrt{85}}{2}$$

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

$$x=\frac{\sqrt{2\sqrt{85}+10}}{2} x=-\frac{\sqrt{2\sqrt{85}+10}}{2}$$

Solve for x (complex solution)

$$x = \frac{\sqrt{2 {(\sqrt{85} + 5)}}}{2} \approx 2.666415614$$

$$x = -\frac{\sqrt{2 {(\sqrt{85} + 5)}}}{2} \approx -2.666415614$$

$$x=-\frac{i\sqrt{2\left(\sqrt{85}-5\right)}}{2}\approx -0-1.4525055i$$

$$x=\frac{i\sqrt{2\left(\sqrt{85}-5\right)}}{2}\approx 1.4525055i$$