Solve x^8-15x^4-16=0

$${ x }^{ 8 } -15 { x }^{ 4 } -16 = 0$$

Solve for x (complex solution)

$$x\in 2,-2,2i,-2i,\sqrt{2}\left(-\frac{1}{2}-\frac{1}{2}i\right),\sqrt{2}\left(\frac{1}{2}-\frac{1}{2}i\right),\sqrt{2}\left(-\frac{1}{2}+\frac{1}{2}i\right),\sqrt{2}\left(\frac{1}{2}+\frac{1}{2}i\right)$$

Steps Using the Quadratic Formula

Substitute t for x^{4}.

$$t^{2}-15t-16=0$$

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, -15 for b, and -16 for c in the quadratic formula.

$$t=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 1\left(-16\right)}}{2}$$

Do the calculations.

$$t=\frac{15±17}{2}$$

Solve the equation t=\frac{15±17}{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.

$$x=-2 x=-2i x=2i x=2 x=\sqrt{2}\left(-\frac{1}{2}-\frac{1}{2}i\right) x=\sqrt{2}\left(\frac{1}{2}+\frac{1}{2}i\right) x=\sqrt{2}\left(-\frac{1}{2}+\frac{1}{2}i\right) x=\sqrt{2}\left(\frac{1}{2}-\frac{1}{2}i\right)$$

Solve for x

$$x=-2$$

$$x=2$$

Steps Using the Quadratic Formula

Substitute t for x^{4}.

$$t^{2}-15t-16=0$$

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, -15 for b, and -16 for c in the quadratic formula.

$$t=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 1\left(-16\right)}}{2}$$

Do the calculations.

$$t=\frac{15±17}{2}$$

Solve the equation t=\frac{15±17}{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.

$$x=2 x=-2$$