Solve x=4x^2-x

$$x = 4 x ^ { 2 } - x$$

Solve for x

$$x=\frac{1}{2}=0.5$$

$$x=0$$

Steps Using Factoring

Subtract 4x^{2} from both sides.

$$x-4x^{2}=-x$$

Add x to both sides.

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

Combine x and x to get 2x.

$$2x-4x^{2}=0$$

Factor out x.

$$x\left(2-4x\right)=0$$

To find equation solutions, solve x=0 and 2-4x=0.

$$x=0 x=\frac{1}{2}$$

Steps Using the Quadratic Formula

Subtract 4x^{2} from both sides.

$$x-4x^{2}=-x$$

Add x to both sides.

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

Combine x and x to get 2x.

$$2x-4x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

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

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

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

Take the square root of 2^{2}.

$$x=\frac{-2±2}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-2±2}{-8}$$

Now solve the equation x=\frac{-2±2}{-8} when ± is plus. Add -2 to 2.

$$x=\frac{0}{-8}$$

Divide 0 by -8.

$$x=0$$

Now solve the equation x=\frac{-2±2}{-8} when ± is minus. Subtract 2 from -2.

$$x=-\frac{4}{-8}$$

Reduce the fraction \frac{-4}{-8} to lowest terms by extracting and canceling out 4.

$$x=\frac{1}{2}$$

The equation is now solved.

$$x=0 x=\frac{1}{2}$$

Steps for Completing the Square

Subtract 4x^{2} from both sides.

$$x-4x^{2}=-x$$

Add x to both sides.

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

Combine x and x to get 2x.

$$2x-4x^{2}=0$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

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

Divide both sides by -4.

$$\frac{-4x^{2}+2x}{-4}=\frac{0}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{2}{-4}x=\frac{0}{-4}$$

Reduce the fraction \frac{2}{-4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{1}{2}x=\frac{0}{-4}$$

Divide 0 by -4.

$$x^{2}-\frac{1}{2}x=0$$

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{1}{16}$$

Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{1}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Simplify.

$$x-\frac{1}{4}=\frac{1}{4} x-\frac{1}{4}=-\frac{1}{4}$$

Add \frac{1}{4} to both sides of the equation.

$$x=\frac{1}{2} x=0$$