Solve 16x^4+56x^2+77=0

$$16 x ^ { 4 } + 56 x ^ { 2 } + 77 = 0$$

Solve for x (complex solution)

$$x=\frac{\sqrt[4]{77}e^{\frac{\left(\arctan(\frac{2\sqrt{7}}{7})+\pi \right)i}{2}}}{2}\approx -0.471031367+1.404233082i$$

$$x=\frac{\sqrt[4]{77}e^{\frac{\arctan(\frac{2\sqrt{7}}{7})i+3\pi i}{2}}}{2}\approx 0.471031367-1.404233082i$$

$$x=\frac{\sqrt[4]{77}e^{\frac{-\arctan(\frac{2\sqrt{7}}{7})i+3\pi i}{2}}}{2}\approx -0.471031367-1.404233082i$$

$$x=\frac{\sqrt[4]{77}e^{\frac{-\arctan(\frac{2\sqrt{7}}{7})i+\pi i}{2}}}{2}\approx 0.471031367+1.404233082i$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$16t^{2}+56t+77=0$$

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 16 for a, 56 for b, and 77 for c in the quadratic formula.

$$t=\frac{-56±\sqrt{56^{2}-4\times 16\times 77}}{2\times 16}$$

Do the calculations.

$$t=\frac{-56±\sqrt{-1792}}{32}$$

Solve the equation t=\frac{-56±\sqrt{-1792}}{32} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{7}i}{2}-\frac{7}{4} t=-\frac{\sqrt{7}i}{2}-\frac{7}{4}$$

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

$$x=\frac{\sqrt[4]{77}e^{\frac{-\arctan(\frac{2\sqrt{7}}{7})i+3\pi i}{2}}}{2} x=\frac{\sqrt[4]{77}e^{\frac{-\arctan(\frac{2\sqrt{7}}{7})i+\pi i}{2}}}{2} x=\frac{\sqrt[4]{77}e^{\frac{\arctan(\frac{2\sqrt{7}}{7})i+3\pi i}{2}}}{2} x=\frac{\sqrt[4]{77}e^{\frac{\left(\arctan(\frac{2\sqrt{7}}{7})+\pi \right)i}{2}}}{2}$$