Solve 127x^4-277x^2+492=0

$$127 { x }^{ 4 } -277 { x }^{ 2 } +492=0$$

Solve for x (complex solution)

$$x=\frac{\sqrt[4]{1007804436}e^{\frac{-\arctan(\frac{\sqrt{173207}}{277})i+2\pi i}{2}}}{127}\approx -1.236689762+0.662457837i$$

$$x=\frac{\sqrt[4]{1007804436}e^{-\frac{\arctan(\frac{\sqrt{173207}}{277})i}{2}}}{127}\approx 1.236689762-0.662457837i$$

$$x=\frac{\sqrt[4]{1007804436}e^{\frac{\arctan(\frac{\sqrt{173207}}{277})i+2\pi i}{2}}}{127}\approx -1.236689762-0.662457837i$$

$$x=\frac{\sqrt[4]{1007804436}e^{\frac{\arctan(\frac{\sqrt{173207}}{277})i}{2}}}{127}\approx 1.236689762+0.662457837i$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$127t^{2}-277t+492=0$$

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 127 for a, -277 for b, and 492 for c in the quadratic formula.

$$t=\frac{-\left(-277\right)±\sqrt{\left(-277\right)^{2}-4\times 127\times 492}}{2\times 127}$$

Do the calculations.

$$t=\frac{277±\sqrt{-173207}}{254}$$

Solve the equation t=\frac{277±\sqrt{-173207}}{254} when ± is plus and when ± is minus.

$$t=\frac{277+\sqrt{173207}i}{254} t=\frac{-\sqrt{173207}i+277}{254}$$

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

$$x=\frac{\sqrt[4]{1007804436}e^{\frac{\arctan(\frac{\sqrt{173207}}{277})i+2\pi i}{2}}}{127} x=\frac{\sqrt[4]{1007804436}e^{\frac{\arctan(\frac{\sqrt{173207}}{277})i}{2}}}{127} x=\frac{\sqrt[4]{1007804436}e^{-\frac{\arctan(\frac{\sqrt{173207}}{277})i}{2}}}{127} x=\frac{\sqrt[4]{1007804436}e^{\frac{-\arctan(\frac{\sqrt{173207}}{277})i+2\pi i}{2}}}{127}$$