Solve 0==x(24-x)

$$0 = = x ( 24 - x )$$

Solve for x

$$x=24$$

$$x=0$$

Steps Using Factoring

Use the distributive property to multiply x by 24-x.

$$0=24x-x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$24x-x^{2}=0$$

Factor out x.

$$x\left(24-x\right)=0$$

To find equation solutions, solve x=0 and 24-x=0.

$$x=0 x=24$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 24-x.

$$0=24x-x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$24x-x^{2}=0$$

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.

$$-x^{2}+24x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 24 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}}}{2\left(-1\right)}$$

Take the square root of 24^{2}.

$$x=\frac{-24±24}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-24±24}{-2}$$

Now solve the equation x=\frac{-24±24}{-2} when ± is plus. Add -24 to 24.

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

Divide 0 by -2.

$$x=0$$

Now solve the equation x=\frac{-24±24}{-2} when ± is minus. Subtract 24 from -24.

$$x=-\frac{48}{-2}$$

Divide -48 by -2.

$$x=24$$

The equation is now solved.

$$x=0 x=24$$

Steps for Completing the Square

Use the distributive property to multiply x by 24-x.

$$0=24x-x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$24x-x^{2}=0$$

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.

$$-x^{2}+24x=0$$

Divide both sides by -1.

$$\frac{-x^{2}+24x}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{24}{-1}x=\frac{0}{-1}$$

Divide 24 by -1.

$$x^{2}-24x=\frac{0}{-1}$$

Divide 0 by -1.

$$x^{2}-24x=0$$

Divide -24, the coefficient of the x term, by 2 to get -12. Then add the square of -12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-24x+\left(-12\right)^{2}=\left(-12\right)^{2}$$

Square -12.

$$x^{2}-24x+144=144$$

Factor x^{2}-24x+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-12\right)^{2}=144$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-12\right)^{2}}=\sqrt{144}$$

Simplify.

$$x-12=12 x-12=-12$$

Add 12 to both sides of the equation.

$$x=24 x=0$$