Solve 4x=x(x-1)

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

Solve for x

$$x=5$$

$$x=0$$

Steps Using Factoring

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

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

Subtract x^{2} from both sides.

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

Add x to both sides.

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

Combine 4x and x to get 5x.

$$5x-x^{2}=0$$

Factor out x.

$$x\left(5-x\right)=0$$

To find equation solutions, solve x=0 and 5-x=0.

$$x=0 x=5$$

Steps Using the Quadratic Formula

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

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

Subtract x^{2} from both sides.

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

Add x to both sides.

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

Combine 4x and x to get 5x.

$$5x-x^{2}=0$$

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 5 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}}}{2\left(-1\right)}$$

Take the square root of 5^{2}.

$$x=\frac{-5±5}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-5±5}{-2}$$

Now solve the equation x=\frac{-5±5}{-2} when ± is plus. Add -5 to 5.

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

Divide 0 by -2.

$$x=0$$

Now solve the equation x=\frac{-5±5}{-2} when ± is minus. Subtract 5 from -5.

$$x=-\frac{10}{-2}$$

Divide -10 by -2.

$$x=5$$

The equation is now solved.

$$x=0 x=5$$

Steps for Completing the Square

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

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

Subtract x^{2} from both sides.

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

Add x to both sides.

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

Combine 4x and x to get 5x.

$$5x-x^{2}=0$$

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.

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

Divide both sides by -1.

$$\frac{-x^{2}+5x}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{5}{-1}x=\frac{0}{-1}$$

Divide 5 by -1.

$$x^{2}-5x=\frac{0}{-1}$$

Divide 0 by -1.

$$x^{2}-5x=0$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=\frac{25}{4}$$

Factor x^{2}-5x+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{5}{2} x-\frac{5}{2}=-\frac{5}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=5 x=0$$