Solve x^4+x^2=6

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

Solve for x (complex solution)

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

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

$$x=-\sqrt{3}i\approx -0-1.732050808i$$

$$x=\sqrt{3}i\approx 1.732050808i$$

Steps Using the Quadratic Formula

Subtract 6 from both sides.

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

Substitute t for x^{2}.

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

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

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

Do the calculations.

$$t=\frac{-1±5}{2}$$

Solve the equation t=\frac{-1±5}{2} when ± is plus and when ± is minus.

$$t=2 t=-3$$

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

$$x=-\sqrt{2} x=\sqrt{2} x=-\sqrt{3}i x=\sqrt{3}i$$

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.

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

Substitute t for x^{2}.

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

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

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

Do the calculations.

$$t=\frac{-1±5}{2}$$

Solve the equation t=\frac{-1±5}{2} when ± is plus and when ± is minus.

$$t=2 t=-3$$

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

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