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

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

Solve for x

$$x=-6$$

$$x=8$$

Steps Using Factoring

Multiply both sides by 2, the reciprocal of \frac{1}{2}.

$$24\times 2=x^{2}-2x$$

Multiply 24 and 2 to get 48.

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

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

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

Subtract 48 from both sides.

$$x^{2}-2x-48=0$$

To solve the equation, factor x^{2}-2x-48 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-2 ab=-48$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -48.

$$1,-48 2,-24 3,-16 4,-12 6,-8$$

Calculate the sum for each pair.

$$1-48=-47 2-24=-22 3-16=-13 4-12=-8 6-8=-2$$

The solution is the pair that gives sum -2.

$$a=-8 b=6$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-8\right)\left(x+6\right)$$

To find equation solutions, solve x-8=0 and x+6=0.

$$x=8 x=-6$$

Steps Using Factoring By Grouping

Multiply both sides by 2, the reciprocal of \frac{1}{2}.

$$24\times 2=x^{2}-2x$$

Multiply 24 and 2 to get 48.

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

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

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

Subtract 48 from both sides.

$$x^{2}-2x-48=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-48. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-48\right)=-48$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -48.

$$1,-48 2,-24 3,-16 4,-12 6,-8$$

Calculate the sum for each pair.

$$1-48=-47 2-24=-22 3-16=-13 4-12=-8 6-8=-2$$

The solution is the pair that gives sum -2.

$$a=-8 b=6$$

Rewrite x^{2}-2x-48 as \left(x^{2}-8x\right)+\left(6x-48\right).

$$\left(x^{2}-8x\right)+\left(6x-48\right)$$

Factor out x in the first and 6 in the second group.

$$x\left(x-8\right)+6\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(x+6\right)$$

To find equation solutions, solve x-8=0 and x+6=0.

$$x=8 x=-6$$

Steps Using the Quadratic Formula

Multiply both sides by 2, the reciprocal of \frac{1}{2}.

$$24\times 2=x^{2}-2x$$

Multiply 24 and 2 to get 48.

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

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

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

Subtract 48 from both sides.

$$x^{2}-2x-48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-48\right)}}{2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-48\right)}}{2}$$

Multiply -4 times -48.

$$x=\frac{-\left(-2\right)±\sqrt{4+192}}{2}$$

Add 4 to 192.

$$x=\frac{-\left(-2\right)±\sqrt{196}}{2}$$

Take the square root of 196.

$$x=\frac{-\left(-2\right)±14}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±14}{2}$$

Now solve the equation x=\frac{2±14}{2} when ± is plus. Add 2 to 14.

$$x=\frac{16}{2}$$

Divide 16 by 2.

$$x=8$$

Now solve the equation x=\frac{2±14}{2} when ± is minus. Subtract 14 from 2.

$$x=-\frac{12}{2}$$

Divide -12 by 2.

$$x=-6$$

The equation is now solved.

$$x=8 x=-6$$

Steps for Completing the Square

Multiply both sides by 2, the reciprocal of \frac{1}{2}.

$$24\times 2=x^{2}-2x$$

Multiply 24 and 2 to get 48.

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

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

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

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.

$$x=8 x=-6$$