Solve x^8-17x^4+16=0

$${ x }^{ 8 } -17 { x }^{ 4 } +16 = 0$$

Solve for x (complex solution)

$$x\in 2,-2,2i,-2i,-1,1,i,-i$$

Steps Using the Quadratic Formula

Substitute t for x^{4}.

$$t^{2}-17t+16=0$$

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, -17 for b, and 16 for c in the quadratic formula.

$$t=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\times 1\times 16}}{2}$$

Do the calculations.

$$t=\frac{17±15}{2}$$

Solve the equation t=\frac{17±15}{2} when ± is plus and when ± is minus.

$$t=16 t=1$$

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

$$x=-2 x=-2i x=2i x=2 x=-1 x=-i x=i x=1$$

Solve for x

$$x=-2$$

$$x=2$$

$$x=1$$

$$x=-1$$

Steps Using the Quadratic Formula

Substitute t for x^{4}.

$$t^{2}-17t+16=0$$

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, -17 for b, and 16 for c in the quadratic formula.

$$t=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\times 1\times 16}}{2}$$

Do the calculations.

$$t=\frac{17±15}{2}$$

Solve the equation t=\frac{17±15}{2} when ± is plus and when ± is minus.

$$t=16 t=1$$

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

$$x=2 x=-2 x=1 x=-1$$