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

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

Solve for x (complex solution)

$$x=i\sqrt{\sqrt{7}+1}\approx 1.909385061i$$

$$x=-i\sqrt{\sqrt{7}+1}\approx -0-1.909385061i$$

$$x=-\sqrt{\sqrt{7}-1}\approx -1.282868392$$

$$x=\sqrt{\sqrt{7}-1}\approx 1.282868392$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

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

Subtract 6 from both sides.

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

Substitute t for x^{2}.

$$t^{2}+2t-6=0$$

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, 2 for b, and -6 for c in the quadratic formula.

$$t=\frac{-2±\sqrt{2^{2}-4\times 1\left(-6\right)}}{2}$$

Do the calculations.

$$t=\frac{-2±2\sqrt{7}}{2}$$

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

$$t=\sqrt{7}-1 t=-\sqrt{7}-1$$

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

$$x=-\sqrt{\sqrt{7}-1} x=\sqrt{\sqrt{7}-1} x=-i\sqrt{\sqrt{7}+1} x=i\sqrt{\sqrt{7}+1}$$

Solve for x

$$x=-\sqrt{\sqrt{7}-1}\approx -1.282868392$$

$$x=\sqrt{\sqrt{7}-1}\approx 1.282868392$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

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

Subtract 6 from both sides.

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

Substitute t for x^{2}.

$$t^{2}+2t-6=0$$

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, 2 for b, and -6 for c in the quadratic formula.

$$t=\frac{-2±\sqrt{2^{2}-4\times 1\left(-6\right)}}{2}$$

Do the calculations.

$$t=\frac{-2±2\sqrt{7}}{2}$$

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

$$t=\sqrt{7}-1 t=-\sqrt{7}-1$$

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

$$x=\sqrt{\sqrt{7}-1} x=-\sqrt{\sqrt{7}-1}$$