Solve x^4-13x^2+112=0

$$x ^ { 4 } - 13 x ^ { 2 } + 112 = 0$$

Solve for x (complex solution)

$$x=2\sqrt[4]{7}e^{\frac{-\arctan(\frac{3\sqrt{31}}{13})i+2\pi i}{2}}\approx -2.922584921+1.428811612i$$

$$x=2\sqrt[4]{7}e^{-\frac{\arctan(\frac{3\sqrt{31}}{13})i}{2}}\approx 2.922584921-1.428811612i$$

$$x=2\sqrt[4]{7}e^{\frac{\arctan(\frac{3\sqrt{31}}{13})i+2\pi i}{2}}\approx -2.922584921-1.428811612i$$

$$x=2\sqrt[4]{7}e^{\frac{\arctan(\frac{3\sqrt{31}}{13})i}{2}}\approx 2.922584921+1.428811612i$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-13t+112=0$$

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, -13 for b, and 112 for c in the quadratic formula.

$$t=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 1\times 112}}{2}$$

Do the calculations.

$$t=\frac{13±\sqrt{-279}}{2}$$

Solve the equation t=\frac{13±\sqrt{-279}}{2} when ± is plus and when ± is minus.

$$t=\frac{13+3\sqrt{31}i}{2} t=\frac{-3\sqrt{31}i+13}{2}$$

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

$$x=2\sqrt[4]{7}e^{\frac{\arctan(\frac{3\sqrt{31}}{13})i+2\pi i}{2}} x=2\sqrt[4]{7}e^{\frac{\arctan(\frac{3\sqrt{31}}{13})i}{2}} x=2\sqrt[4]{7}e^{-\frac{\arctan(\frac{3\sqrt{31}}{13})i}{2}} x=2\sqrt[4]{7}e^{\frac{-\arctan(\frac{3\sqrt{31}}{13})i+2\pi i}{2}}$$