Solve x^8-5x^4+4=0

$$x ^ { 8 } - 5 x ^ { 4 } + 4 = 0$$

Solve for x (complex solution)

$$x\in \sqrt{2},-\sqrt{2}i,-\sqrt{2},\sqrt{2}i,-1,1,i,-i$$

Steps Using the Quadratic Formula

Substitute t for x^{4}.

$$t^{2}-5t+4=0$$

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, -5 for b, and 4 for c in the quadratic formula.

$$t=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 1\times 4}}{2}$$

Do the calculations.

$$t=\frac{5±3}{2}$$

Solve the equation t=\frac{5±3}{2} when ± is plus and when ± is minus.

$$t=4 t=1$$

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

$$x=-\sqrt{2}i x=\sqrt{2}i x=-\sqrt{2} x=\sqrt{2} x=-1 x=-i x=i x=1$$

Solve for x

$$x=-\sqrt{2}\approx -1.414213562$$

$$x=\sqrt{2}\approx 1.414213562$$

$$x=1$$

$$x=-1$$

Steps Using the Quadratic Formula

Substitute t for x^{4}.

$$t^{2}-5t+4=0$$

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, -5 for b, and 4 for c in the quadratic formula.

$$t=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 1\times 4}}{2}$$

Do the calculations.

$$t=\frac{5±3}{2}$$

Solve the equation t=\frac{5±3}{2} when ± is plus and when ± is minus.

$$t=4 t=1$$

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

$$x=\sqrt{2} x=-\sqrt{2} x=1 x=-1$$