Solve 18x^4+72x^2+169=0

$$18 x ^ { 4 } + 72 x ^ { 2 } + 169 = 0$$

Solve for x (complex solution)

$$x=\frac{\sqrt[4]{12168}e^{\frac{\left(\arctan(\frac{\sqrt{194}}{12})+\pi \right)i}{2}}}{6}\approx -0.729427647+1.59124627i$$

$$x=\frac{\sqrt[4]{12168}e^{\frac{\arctan(\frac{\sqrt{194}}{12})i+3\pi i}{2}}}{6}\approx 0.729427647-1.59124627i$$

$$x=\frac{\sqrt[4]{12168}e^{\frac{-\arctan(\frac{\sqrt{194}}{12})i+3\pi i}{2}}}{6}\approx -0.729427647-1.59124627i$$

$$x=\frac{\sqrt[4]{12168}e^{\frac{-\arctan(\frac{\sqrt{194}}{12})i+\pi i}{2}}}{6}\approx 0.729427647+1.59124627i$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$18t^{2}+72t+169=0$$

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 18 for a, 72 for b, and 169 for c in the quadratic formula.

$$t=\frac{-72±\sqrt{72^{2}-4\times 18\times 169}}{2\times 18}$$

Do the calculations.

$$t=\frac{-72±\sqrt{-6984}}{36}$$

Solve the equation t=\frac{-72±\sqrt{-6984}}{36} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{194}i}{6}-2 t=-\frac{\sqrt{194}i}{6}-2$$

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

$$x=\frac{\sqrt[4]{12168}e^{\frac{-\arctan(\frac{\sqrt{194}}{12})i+3\pi i}{2}}}{6} x=\frac{\sqrt[4]{12168}e^{\frac{-\arctan(\frac{\sqrt{194}}{12})i+\pi i}{2}}}{6} x=\frac{\sqrt[4]{12168}e^{\frac{\arctan(\frac{\sqrt{194}}{12})i+3\pi i}{2}}}{6} x=\frac{\sqrt[4]{12168}e^{\frac{\left(\arctan(\frac{\sqrt{194}}{12})+\pi \right)i}{2}}}{6}$$