Solve 10x^2+x^4=8

$$10 x ^ { 2 } + x ^ { 4 } = 8$$

Solve for x (complex solution)

$$x=i\sqrt{\sqrt{33}+5}\approx 3.277889969i$$

$$x=-i\sqrt{\sqrt{33}+5}\approx -0-3.277889969i$$

$$x=-\sqrt{\sqrt{33}-5}\approx -0.862880436$$

$$x=\sqrt{\sqrt{33}-5}\approx 0.862880436$$

Steps Using the Quadratic Formula

Subtract 8 from both sides.

$$10x^{2}+x^{4}-8=0$$

Substitute t for x^{2}.

$$t^{2}+10t-8=0$$

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 -8 for c in the quadratic formula.

$$t=\frac{-10±\sqrt{10^{2}-4\times 1\left(-8\right)}}{2}$$

Do the calculations.

$$t=\frac{-10±2\sqrt{33}}{2}$$

Solve the equation t=\frac{-10±2\sqrt{33}}{2} when ± is plus and when ± is minus.

$$t=\sqrt{33}-5 t=-\sqrt{33}-5$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for each t.

$$x=-\sqrt{\sqrt{33}-5} x=\sqrt{\sqrt{33}-5} x=-i\sqrt{\sqrt{33}+5} x=i\sqrt{\sqrt{33}+5}$$

Solve for x

$$x=-\sqrt{\sqrt{33}-5}\approx -0.862880436$$

$$x=\sqrt{\sqrt{33}-5}\approx 0.862880436$$

Steps Using the Quadratic Formula

Subtract 8 from both sides.

$$10x^{2}+x^{4}-8=0$$

Substitute t for x^{2}.

$$t^{2}+10t-8=0$$

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 -8 for c in the quadratic formula.

$$t=\frac{-10±\sqrt{10^{2}-4\times 1\left(-8\right)}}{2}$$

Do the calculations.

$$t=\frac{-10±2\sqrt{33}}{2}$$

Solve the equation t=\frac{-10±2\sqrt{33}}{2} when ± is plus and when ± is minus.

$$t=\sqrt{33}-5 t=-\sqrt{33}-5$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for positive t.

$$x=\sqrt{\sqrt{33}-5} x=-\sqrt{\sqrt{33}-5}$$