Solve y^4-3y^2+4=0

$$y ^ { 4 } - 3 y ^ { 2 } + 4 = 0$$

Solve for y

$$y=\sqrt{2}e^{\frac{-\arctan(\frac{\sqrt{7}}{3})i+2\pi i}{2}}\approx -1.322875656+0.5i$$

$$y=\sqrt{2}e^{-\frac{\arctan(\frac{\sqrt{7}}{3})i}{2}}\approx 1.322875656-0.5i$$

$$y=\sqrt{2}e^{\frac{\arctan(\frac{\sqrt{7}}{3})i+2\pi i}{2}}\approx -1.322875656-0.5i$$

$$y=\sqrt{2}e^{\frac{\arctan(\frac{\sqrt{7}}{3})i}{2}}\approx 1.322875656+0.5i$$

Steps Using the Quadratic Formula

Substitute t for y^{2}.

$$t^{2}-3t+4=0$$

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, -3 for b, and 4 for c in the quadratic formula.

$$t=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 1\times 4}}{2}$$

Do the calculations.

$$t=\frac{3±\sqrt{-7}}{2}$$

Solve the equation t=\frac{3±\sqrt{-7}}{2} when ± is plus and when ± is minus.

$$t=\frac{3+\sqrt{7}i}{2} t=\frac{-\sqrt{7}i+3}{2}$$

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

$$y=\sqrt{2}e^{\frac{\arctan(\frac{\sqrt{7}}{3})i+2\pi i}{2}} y=\sqrt{2}e^{\frac{\arctan(\frac{\sqrt{7}}{3})i}{2}} y=\sqrt{2}e^{-\frac{\arctan(\frac{\sqrt{7}}{3})i}{2}} y=\sqrt{2}e^{\frac{-\arctan(\frac{\sqrt{7}}{3})i+2\pi i}{2}}$$