Solve x=2x^2+x-3

$$x=2 { x }^{ 2 } +x-3$$

Solve for x

$$x = \frac{\sqrt{6}}{2} \approx 1.224744871$$

$$x = -\frac{\sqrt{6}}{2} \approx -1.224744871$$

Steps by Finding Square Root

Subtract 2x^{2} from both sides.

$$x-2x^{2}=x-3$$

Subtract x from both sides.

$$x-2x^{2}-x=-3$$

Combine x and -x to get 0.

$$-2x^{2}=-3$$

Divide both sides by -2.

$$x^{2}=\frac{-3}{-2}$$

Fraction \frac{-3}{-2} can be simplified to \frac{3}{2} by removing the negative sign from both the numerator and the denominator.

$$x^{2}=\frac{3}{2}$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{6}}{2} x=-\frac{\sqrt{6}}{2}$$

Steps Using the Quadratic Formula

Subtract 2x^{2} from both sides.

$$x-2x^{2}=x-3$$

Subtract x from both sides.

$$x-2x^{2}-x=-3$$

Combine x and -x to get 0.

$$-2x^{2}=-3$$

Add 3 to both sides.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 0 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-2\right)\times 3}}{2\left(-2\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-2\right)\times 3}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{0±\sqrt{8\times 3}}{2\left(-2\right)}$$

Multiply 8 times 3.

$$x=\frac{0±\sqrt{24}}{2\left(-2\right)}$$

Take the square root of 24.

$$x=\frac{0±2\sqrt{6}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{0±2\sqrt{6}}{-4}$$

Now solve the equation x=\frac{0±2\sqrt{6}}{-4} when ± is plus.

$$x=-\frac{\sqrt{6}}{2}$$

Now solve the equation x=\frac{0±2\sqrt{6}}{-4} when ± is minus.

$$x=\frac{\sqrt{6}}{2}$$

The equation is now solved.

$$x=-\frac{\sqrt{6}}{2} x=\frac{\sqrt{6}}{2}$$