Solve left(.5xright)^4+2x^2-1=0

$${ \left(.5x \right) }^{ 4 } +2 { x }^{ 2 } -1 = 0$$

Solve for x (complex solution)

$$x=-i\times 2\sqrt{\sqrt{17}+4}\approx -0-5.700212496i$$

$$x=i\times 2\sqrt{\sqrt{17}+4}\approx 5.700212496i$$

$$x=2\sqrt{\sqrt{17}-4}\approx 0.701728226$$

$$x=-2\sqrt{\sqrt{17}-4}\approx -0.701728226$$

Steps Using the Quadratic Formula

Expand \left(0.5x\right)^{4}.

$$0.5^{4}x^{4}+2x^{2}-1=0$$

Calculate 0.5 to the power of 4 and get 0.0625.

$$0.0625x^{4}+2x^{2}-1=0$$

Substitute t for x^{2}.

$$0.0625t^{2}+2t-1=0$$

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 0.0625 for a, 2 for b, and -1 for c in the quadratic formula.

$$t=\frac{-2±\sqrt{2^{2}-4\times 0.0625\left(-1\right)}}{0.0625\times 2}$$

Do the calculations.

$$t=\frac{-2±\frac{1}{2}\sqrt{17}}{0.125}$$

Solve the equation t=\frac{-2±\frac{1}{2}\sqrt{17}}{0.125} when ± is plus and when ± is minus.

$$t=4\sqrt{17}-16 t=-4\sqrt{17}-16$$

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

$$x=-2\sqrt{\sqrt{17}-4} x=2\sqrt{\sqrt{17}-4} x=-2i\sqrt{\sqrt{17}+4} x=2i\sqrt{\sqrt{17}+4}$$

Solve for x

$$x=2\sqrt{\sqrt{17}-4}\approx 0.701728226$$

$$x=-2\sqrt{\sqrt{17}-4}\approx -0.701728226$$

Steps Using the Quadratic Formula

Expand \left(0.5x\right)^{4}.

$$0.5^{4}x^{4}+2x^{2}-1=0$$

Calculate 0.5 to the power of 4 and get 0.0625.

$$0.0625x^{4}+2x^{2}-1=0$$

Substitute t for x^{2}.

$$0.0625t^{2}+2t-1=0$$

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 0.0625 for a, 2 for b, and -1 for c in the quadratic formula.

$$t=\frac{-2±\sqrt{2^{2}-4\times 0.0625\left(-1\right)}}{0.0625\times 2}$$

Do the calculations.

$$t=\frac{-2±\frac{1}{2}\sqrt{17}}{0.125}$$

Solve the equation t=\frac{-2±\frac{1}{2}\sqrt{17}}{0.125} when ± is plus and when ± is minus.

$$t=4\sqrt{17}-16 t=-4\sqrt{17}-16$$

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

$$x=2\sqrt{\sqrt{17}-4} x=-2\sqrt{\sqrt{17}-4}$$