Solve x^4+x^2=20

$$x ^ { 4 } + x ^ { 2 } = 20$$

Solve for x (complex solution)

$$x=2$$

$$x=-2$$

$$x=-\sqrt{5}i\approx -0-2.236067977i$$

$$x=\sqrt{5}i\approx 2.236067977i$$

Steps Using the Quadratic Formula

Subtract 20 from both sides.

$$x^{4}+x^{2}-20=0$$

Substitute t for x^{2}.

$$t^{2}+t-20=0$$

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 -20 for c in the quadratic formula.

$$t=\frac{-1±\sqrt{1^{2}-4\times 1\left(-20\right)}}{2}$$

Do the calculations.

$$t=\frac{-1±9}{2}$$

Solve the equation t=\frac{-1±9}{2} when ± is plus and when ± is minus.

$$t=4 t=-5$$

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

$$x=-2 x=2 x=-\sqrt{5}i x=\sqrt{5}i$$

Solve for x

$$x=-2$$

$$x=2$$

Steps Using the Quadratic Formula

Subtract 20 from both sides.

$$x^{4}+x^{2}-20=0$$

Substitute t for x^{2}.

$$t^{2}+t-20=0$$

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 -20 for c in the quadratic formula.

$$t=\frac{-1±\sqrt{1^{2}-4\times 1\left(-20\right)}}{2}$$

Do the calculations.

$$t=\frac{-1±9}{2}$$

Solve the equation t=\frac{-1±9}{2} when ± is plus and when ± is minus.

$$t=4 t=-5$$

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

$$x=2 x=-2$$