Solve 24=1/2(x)(x-2)

$$24= \frac{ 1 }{ 2 } (x)(x-2)$$

Solve for x

$$x=-6$$

$$x=8$$

Steps Using the Quadratic Formula

Use the distributive property to multiply \frac{1}{2}x by x-2.

$$24=\frac{1}{2}xx+\frac{1}{2}x\left(-2\right)$$

Multiply x and x to get x^{2}.

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

Multiply \frac{1}{2} and -2 to get \frac{-2}{2}.

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

Divide -2 by 2 to get -1.

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

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

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

Subtract 24 from both sides.

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

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

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times \frac{1}{2}\left(-24\right)}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

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

Multiply -2 times -24.

$$x=\frac{-\left(-1\right)±\sqrt{1+48}}{2\times \frac{1}{2}}$$

Add 1 to 48.

$$x=\frac{-\left(-1\right)±\sqrt{49}}{2\times \frac{1}{2}}$$

Take the square root of 49.

$$x=\frac{-\left(-1\right)±7}{2\times \frac{1}{2}}$$

The opposite of -1 is 1.

$$x=\frac{1±7}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$x=\frac{1±7}{1}$$

Now solve the equation x=\frac{1±7}{1} when ± is plus. Add 1 to 7.

$$x=\frac{8}{1}$$

Divide 8 by 1.

$$x=8$$

Now solve the equation x=\frac{1±7}{1} when ± is minus. Subtract 7 from 1.

$$x=-\frac{6}{1}$$

Divide -6 by 1.

$$x=-6$$

The equation is now solved.

$$x=8 x=-6$$

Steps for Completing the Square

Use the distributive property to multiply \frac{1}{2}x by x-2.

$$24=\frac{1}{2}xx+\frac{1}{2}x\left(-2\right)$$

Multiply x and x to get x^{2}.

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

Multiply \frac{1}{2} and -2 to get \frac{-2}{2}.

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

Divide -2 by 2 to get -1.

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

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

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

Multiply both sides by 2.

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

Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.

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

Divide -1 by \frac{1}{2} by multiplying -1 by the reciprocal of \frac{1}{2}.

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

Divide 24 by \frac{1}{2} by multiplying 24 by the reciprocal of \frac{1}{2}.

$$x^{2}-2x=48$$

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

$$x^{2}-2x+1=48+1$$

Add 48 to 1.

$$x^{2}-2x+1=49$$

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

$$\left(x-1\right)^{2}=49$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{49}$$

Simplify.

$$x-1=7 x-1=-7$$

Add 1 to both sides of the equation.

$$x=8 x=-6$$