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

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

Solve for x

$$x=\frac{\sqrt{2}}{2}\approx 0.707106781$$

$$x=-\frac{\sqrt{2}}{2}\approx -0.707106781$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+1\right)\left(-x^{2}-1\right).

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

Use the distributive property to multiply x-1 by x+1 and combine like terms.

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

Use the distributive property to multiply x^{2}-1 by -x^{2}-1 and combine like terms.

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

Add x^{4} to both sides.

$$x^{4}+x^{2}+x^{4}=1$$

Combine x^{4} and x^{4} to get 2x^{4}.

$$2x^{4}+x^{2}=1$$

Subtract 1 from both sides.

$$2x^{4}+x^{2}-1=0$$

Substitute t for x^{2}.

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

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

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

Do the calculations.

$$t=\frac{-1±3}{4}$$

Solve the equation t=\frac{-1±3}{4} when ± is plus and when ± is minus.

$$t=\frac{1}{2} t=-1$$

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

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