Solve x^4-4x^2-21=0

$${ x }^{ 4 } -4 { x }^{ 2 } -21 = 0$$

Solve for x (complex solution)

$$x=\sqrt{7}\approx 2.645751311$$

$$x=-\sqrt{7}\approx -2.645751311$$

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

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

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-4t-21=0$$

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 -21 for c in the quadratic formula.

$$t=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 1\left(-21\right)}}{2}$$

Do the calculations.

$$t=\frac{4±10}{2}$$

Solve the equation t=\frac{4±10}{2} when ± is plus and when ± is minus.

$$t=7 t=-3$$

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

$$x=-\sqrt{7} x=\sqrt{7} x=-\sqrt{3}i x=\sqrt{3}i$$

Solve for x

$$x=-\sqrt{7}\approx -2.645751311$$

$$x=\sqrt{7}\approx 2.645751311$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-4t-21=0$$

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 -21 for c in the quadratic formula.

$$t=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 1\left(-21\right)}}{2}$$

Do the calculations.

$$t=\frac{4±10}{2}$$

Solve the equation t=\frac{4±10}{2} when ± is plus and when ± is minus.

$$t=7 t=-3$$

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

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