Solve x^4-30x^2-32=0

$$x ^ { 4 } - 30 x ^ { 2 } - 32 = 0$$

Solve for x (complex solution)

$$x=i\sqrt{\sqrt{257}-15}\approx 1.015489804i$$

$$x=-i\sqrt{\sqrt{257}-15}\approx -0-1.015489804i$$

$$x=-\sqrt{\sqrt{257}+15}\approx -5.570567255$$

$$x=\sqrt{\sqrt{257}+15}\approx 5.570567255$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-30t-32=0$$

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, -30 for b, and -32 for c in the quadratic formula.

$$t=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 1\left(-32\right)}}{2}$$

Do the calculations.

$$t=\frac{30±2\sqrt{257}}{2}$$

Solve the equation t=\frac{30±2\sqrt{257}}{2} when ± is plus and when ± is minus.

$$t=\sqrt{257}+15 t=15-\sqrt{257}$$

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

$$x=-\sqrt{\sqrt{257}+15} x=\sqrt{\sqrt{257}+15} x=-i\sqrt{-\left(15-\sqrt{257}\right)} x=i\sqrt{-\left(15-\sqrt{257}\right)}$$

Solve for x

$$x=-\sqrt{\sqrt{257}+15}\approx -5.570567255$$

$$x=\sqrt{\sqrt{257}+15}\approx 5.570567255$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-30t-32=0$$

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, -30 for b, and -32 for c in the quadratic formula.

$$t=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 1\left(-32\right)}}{2}$$

Do the calculations.

$$t=\frac{30±2\sqrt{257}}{2}$$

Solve the equation t=\frac{30±2\sqrt{257}}{2} when ± is plus and when ± is minus.

$$t=\sqrt{257}+15 t=15-\sqrt{257}$$

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

$$x=\sqrt{\sqrt{257}+15} x=-\sqrt{\sqrt{257}+15}$$