Solve x^2(13-x^2)=89

$$x ^ { 2 } ( 13 - x ^ { 2 } ) = 89$$

Solve for x (complex solution)

$$x=\sqrt[4]{89}e^{\frac{\arctan(\frac{\sqrt{187}}{13})i}{2}}\approx 2.822585794+1.21119386i$$

$$x=\sqrt[4]{89}e^{\frac{\arctan(\frac{\sqrt{187}}{13})i+2\pi i}{2}}\approx -2.822585794-1.21119386i$$

$$x=\sqrt[4]{89}e^{-\frac{\arctan(\frac{\sqrt{187}}{13})i}{2}}\approx 2.822585794-1.21119386i$$

$$x=\sqrt[4]{89}e^{\frac{-\arctan(\frac{\sqrt{187}}{13})i+2\pi i}{2}}\approx -2.822585794+1.21119386i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x^{2} by 13-x^{2}.

$$13x^{2}-x^{4}=89$$

Subtract 89 from both sides.

$$13x^{2}-x^{4}-89=0$$

Substitute t for x^{2}.

$$-t^{2}+13t-89=0$$

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 -89 for c in the quadratic formula.

$$t=\frac{-13±\sqrt{13^{2}-4\left(-1\right)\left(-89\right)}}{-2}$$

Do the calculations.

$$t=\frac{-13±\sqrt{-187}}{-2}$$

Solve the equation t=\frac{-13±\sqrt{-187}}{-2} when ± is plus and when ± is minus.

$$t=\frac{-\sqrt{187}i+13}{2} t=\frac{13+\sqrt{187}i}{2}$$

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

$$x=\sqrt[4]{89}e^{-\frac{\arctan(\frac{\sqrt{187}}{13})i}{2}} x=\sqrt[4]{89}e^{\frac{-\arctan(\frac{\sqrt{187}}{13})i+2\pi i}{2}} x=\sqrt[4]{89}e^{\frac{\arctan(\frac{\sqrt{187}}{13})i+2\pi i}{2}} x=\sqrt[4]{89}e^{\frac{\arctan(\frac{\sqrt{187}}{13})i}{2}}$$