Solve x^4-6x^2+10=0

$$x ^ { 4 } - 6 x ^ { 2 } + 10 = 0$$

Solve for x (complex solution)

$$x=\sqrt[4]{10}e^{\frac{-\arctan(\frac{1}{3})i+2\pi i}{2}}\approx -1.755317302+0.284848785i$$

$$x=\sqrt[4]{10}e^{-\frac{\arctan(\frac{1}{3})i}{2}}\approx 1.755317302-0.284848785i$$

$$x=\sqrt[4]{10}e^{\frac{\arctan(\frac{1}{3})i+2\pi i}{2}}\approx -1.755317302-0.284848785i$$

$$x=\sqrt[4]{10}e^{\frac{\arctan(\frac{1}{3})i}{2}}\approx 1.755317302+0.284848785i$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-6t+10=0$$

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, -6 for b, and 10 for c in the quadratic formula.

$$t=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 1\times 10}}{2}$$

Do the calculations.

$$t=\frac{6±\sqrt{-4}}{2}$$

Solve the equation t=\frac{6±\sqrt{-4}}{2} when ± is plus and when ± is minus.

$$t=3+i t=3-i$$

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

$$x=\sqrt[4]{10}e^{\frac{\arctan(\frac{1}{3})i+2\pi i}{2}} x=\sqrt[4]{10}e^{\frac{\arctan(\frac{1}{3})i}{2}} x=\sqrt[4]{10}e^{-\frac{\arctan(\frac{1}{3})i}{2}} x=\sqrt[4]{10}e^{\frac{-\arctan(\frac{1}{3})i+2\pi i}{2}}$$