Solve x^8-77x^4+16=0

$$x ^ { 8 } - 77 x ^ { 4 } + 16 = 0$$

Solve for x (complex solution)

$$x\in \frac{\sqrt{2\left(\sqrt{85}-\sqrt{69}\right)}}{2},-\frac{i\sqrt{2\left(\sqrt{85}-\sqrt{69}\right)}}{2},\frac{i\sqrt{2\left(\sqrt{85}-\sqrt{69}\right)}}{2},-\frac{\sqrt{2\left(\sqrt{85}-\sqrt{69}\right)}}{2},-\frac{i\sqrt{2\left(\sqrt{69}+\sqrt{85}\right)}}{2},\frac{\sqrt{2\left(\sqrt{69}+\sqrt{85}\right)}}{2},\frac{i\sqrt{2\left(\sqrt{69}+\sqrt{85}\right)}}{2},-\frac{\sqrt{2\left(\sqrt{69}+\sqrt{85}\right)}}{2}$$

Steps Using the Quadratic Formula

Substitute t for x^{4}.

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

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

$$t=\frac{-\left(-77\right)±\sqrt{\left(-77\right)^{2}-4\times 1\times 16}}{2}$$

Do the calculations.

$$t=\frac{77±\sqrt{5865}}{2}$$

Solve the equation t=\frac{77±\sqrt{5865}}{2} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{5865}+77}{2} t=\frac{77-\sqrt{5865}}{2}$$

Since x=t^{4}, the solutions are obtained by solving the equation for each t.

$$x=-\frac{i\sqrt{2\sqrt{69}+2\sqrt{85}}}{2} x=-\frac{\sqrt{2\sqrt{69}+2\sqrt{85}}}{2} x=\frac{i\sqrt{2\sqrt{69}+2\sqrt{85}}}{2} x=\frac{\sqrt{2\sqrt{69}+2\sqrt{85}}}{2} x=-\frac{i\sqrt{2\sqrt{85}-2\sqrt{69}}}{2} x=-\frac{\sqrt{2\sqrt{85}-2\sqrt{69}}}{2} x=\frac{i\sqrt{2\sqrt{85}-2\sqrt{69}}}{2} x=\frac{\sqrt{2\sqrt{85}-2\sqrt{69}}}{2}$$

Solve for x

$$x=-\frac{\sqrt{2\left(\sqrt{85}-\sqrt{69}\right)}}{2}\approx -0.675618455$$

$$x=\frac{\sqrt{2\left(\sqrt{85}-\sqrt{69}\right)}}{2}\approx 0.675618455$$

$$x = \frac{\sqrt{2 {(\sqrt{69} + \sqrt{85})}}}{2} \approx 2.960250692$$

$$x = -\frac{\sqrt{2 {(\sqrt{69} + \sqrt{85})}}}{2} \approx -2.960250692$$

Steps Using the Quadratic Formula

Substitute t for x^{4}.

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

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

$$t=\frac{-\left(-77\right)±\sqrt{\left(-77\right)^{2}-4\times 1\times 16}}{2}$$

Do the calculations.

$$t=\frac{77±\sqrt{5865}}{2}$$

Solve the equation t=\frac{77±\sqrt{5865}}{2} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{5865}+77}{2} t=\frac{77-\sqrt{5865}}{2}$$

Since x=t^{4}, the solutions are obtained by evaluating x=±\sqrt[4]{t} for positive t.

$$x=\frac{\sqrt{2\sqrt{69}+2\sqrt{85}}}{2} x=-\frac{\sqrt{2\sqrt{69}+2\sqrt{85}}}{2} x=\frac{\sqrt{2\sqrt{85}-2\sqrt{69}}}{2} x=-\frac{\sqrt{2\sqrt{85}-2\sqrt{69}}}{2}$$