Solve x^2+1/4+1/x^2=3/2

$$\frac { x ^ { 2 } + 1 } { 4 } + \frac { 1 } { x ^ { 2 } } = \frac { 3 } { 2 }$$

Solve for x

$$x=-1$$

$$x=1$$

$$x=2$$

$$x=-2$$

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 4x^{2}, the least common multiple of 4,x^{2},2.

$$x^{2}\left(x^{2}+1\right)+4=6x^{2}$$

Use the distributive property to multiply x^{2} by x^{2}+1.

$$x^{4}+x^{2}+4=6x^{2}$$

Subtract 6x^{2} from both sides.

$$x^{4}+x^{2}+4-6x^{2}=0$$

Combine x^{2} and -6x^{2} to get -5x^{2}.

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

Substitute t for x^{2}.

$$t^{2}-5t+4=0$$

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 4 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{5±3}{2}$$

Solve the equation t=\frac{5±3}{2} when ± is plus and when ± is minus.

$$t=4 t=1$$

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

$$x=2 x=-2 x=1 x=-1$$