Solve -4x(x-1)=2x-2

$$-4x(x-1)=2x-2$$

Solve for x

$$x=1$$

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

Steps Using the Quadratic Formula

Use the distributive property to multiply -4x by x-1.

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

Subtract 2x from both sides.

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

Combine 4x and -2x to get 2x.

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

Add 2 to both sides.

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

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

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

Square 2.

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

Multiply -4 times -4.

$$x=\frac{-2±\sqrt{4+16\times 2}}{2\left(-4\right)}$$

Multiply 16 times 2.

$$x=\frac{-2±\sqrt{4+32}}{2\left(-4\right)}$$

Add 4 to 32.

$$x=\frac{-2±\sqrt{36}}{2\left(-4\right)}$$

Take the square root of 36.

$$x=\frac{-2±6}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-2±6}{-8}$$

Now solve the equation x=\frac{-2±6}{-8} when ± is plus. Add -2 to 6.

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

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

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

Now solve the equation x=\frac{-2±6}{-8} when ± is minus. Subtract 6 from -2.

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

Divide -8 by -8.

$$x=1$$

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply -4x by x-1.

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

Subtract 2x from both sides.

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

Combine 4x and -2x to get 2x.

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

Divide both sides by -4.

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

Dividing by -4 undoes the multiplication by -4.

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

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

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

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

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

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.

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

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

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

Add \frac{1}{2} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{9}{16}$$

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}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{9}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Simplify.

$$x-\frac{1}{4}=\frac{3}{4} x-\frac{1}{4}=-\frac{3}{4}$$

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

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