Solve a^4-15a^2+90=0

$$a ^ { 4 } - 15 a ^ { 2 } + 90 = 0$$

Solve for a

$$a=\sqrt{3}\sqrt[4]{10}e^{\frac{-\arctan(\frac{\sqrt{15}}{5})i+2\pi i}{2}}\approx -2.914346666+0.996702809i$$

$$a=\sqrt{3}\sqrt[4]{10}e^{-\frac{\arctan(\frac{\sqrt{15}}{5})i}{2}}\approx 2.914346666-0.996702809i$$

$$a=\sqrt{3}\sqrt[4]{10}e^{\frac{\arctan(\frac{\sqrt{15}}{5})i+2\pi i}{2}}\approx -2.914346666-0.996702809i$$

$$a=\sqrt{3}\sqrt[4]{10}e^{\frac{\arctan(\frac{\sqrt{15}}{5})i}{2}}\approx 2.914346666+0.996702809i$$

Steps Using the Quadratic Formula

Substitute t for a^{2}.

$$t^{2}-15t+90=0$$

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, -15 for b, and 90 for c in the quadratic formula.

$$t=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 1\times 90}}{2}$$

Do the calculations.

$$t=\frac{15±\sqrt{-135}}{2}$$

Solve the equation t=\frac{15±\sqrt{-135}}{2} when ± is plus and when ± is minus.

$$t=\frac{15+3\sqrt{15}i}{2} t=\frac{-3\sqrt{15}i+15}{2}$$

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

$$a=\sqrt{3}\sqrt[4]{10}e^{\frac{\arctan(\frac{\sqrt{15}}{5})i+2\pi i}{2}} a=\sqrt{3}\sqrt[4]{10}e^{\frac{\arctan(\frac{\sqrt{15}}{5})i}{2}} a=\sqrt{3}\sqrt[4]{10}e^{-\frac{\arctan(\frac{\sqrt{15}}{5})i}{2}} a=\sqrt{3}\sqrt[4]{10}e^{\frac{-\arctan(\frac{\sqrt{15}}{5})i+2\pi i}{2}}$$