Solve x^8-10x^4=-9

$${ x }^{ 8 } -10 { x }^{ 4 } =-9$$

Solve for x (complex solution)

$$x\in \sqrt{3},-\sqrt{3}i,-\sqrt{3},\sqrt{3}i,-1,1,i,-i$$

Steps Using the Quadratic Formula

Add 9 to both sides.

$$x^{8}-10x^{4}+9=0$$

Substitute t for x^{4}.

$$t^{2}-10t+9=0$$

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, -10 for b, and 9 for c in the quadratic formula.

$$t=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 1\times 9}}{2}$$

Do the calculations.

$$t=\frac{10±8}{2}$$

Solve the equation t=\frac{10±8}{2} when ± is plus and when ± is minus.

$$t=9 t=1$$

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

$$x=-\sqrt{3}i x=\sqrt{3}i x=-\sqrt{3} x=\sqrt{3} x=-1 x=-i x=i x=1$$

Solve for x

$$x=-\sqrt{3}\approx -1.732050808$$

$$x=\sqrt{3}\approx 1.732050808$$

$$x=1$$

$$x=-1$$

Steps Using the Quadratic Formula

Add 9 to both sides.

$$x^{8}-10x^{4}+9=0$$

Substitute t for x^{4}.

$$t^{2}-10t+9=0$$

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, -10 for b, and 9 for c in the quadratic formula.

$$t=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 1\times 9}}{2}$$

Do the calculations.

$$t=\frac{10±8}{2}$$

Solve the equation t=\frac{10±8}{2} when ± is plus and when ± is minus.

$$t=9 t=1$$

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

$$x=\sqrt{3} x=-\sqrt{3} x=1 x=-1$$