Solve 2+4x^2-x^4=6

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

Solve for x

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

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

Steps Using the Quadratic Formula

Subtract 6 from both sides.

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

Subtract 6 from 2 to get -4.

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

Substitute t for x^{2}.

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

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, 4 for b, and -4 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{-4±0}{-2}$$

Solutions are the same.

$$t=2$$

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

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