Solve 5x^4-3x^2+2x=0

$$5 x ^ { 4 } - 3 x ^ { 2 } + 2 x = 0$$

Solve for x (complex solution)

$$x=-1$$

$$x=0$$

$$x=-\frac{\sqrt{15}i}{10}+\frac{1}{2}\approx 0.5-0.387298335i$$

$$x=\frac{\sqrt{15}i}{10}+\frac{1}{2}\approx 0.5+0.387298335i$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$5t^{2}-3t+2=0$$

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 5 for a, -3 for b, and 2 for c in the quadratic formula.

$$t=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 5\times 2}}{2\times 5}$$

Do the calculations.

$$t=\frac{3±\sqrt{-31}}{10}$$

Solve the equation t=\frac{3±\sqrt{-31}}{10} when ± is plus and when ± is minus.

$$t=\frac{3+\sqrt{31}i}{10} t=\frac{-\sqrt{31}i+3}{10}$$

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

$$x=\frac{\sqrt[4]{250}e^{\frac{\arctan(\frac{\sqrt{31}}{3})i+2\pi i}{2}}}{5} x=\frac{\sqrt[4]{250}e^{\frac{\arctan(\frac{\sqrt{31}}{3})i}{2}}}{5} x=\frac{\sqrt[4]{250}e^{-\frac{\arctan(\frac{\sqrt{31}}{3})i}{2}}}{5} x=\frac{\sqrt[4]{250}e^{\frac{-\arctan(\frac{\sqrt{31}}{3})i+2\pi i}{2}}}{5}$$

Solve for x

$$x=-1$$

$$x=0$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$5t^{2}-3t+2=0$$

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 5 for a, -3 for b, and 2 for c in the quadratic formula.

$$t=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 5\times 2}}{2\times 5}$$

Do the calculations.

$$t=\frac{3±\sqrt{-31}}{10}$$

Since the square root of a negative number is not defined in the real field, there are no solutions.

$$t\in \emptyset$$

Since t=x^{2}, the original equation does not have any solutions.

$$x\in \emptyset$$