Solve 12x^2-24x=-(x-2)

$$12 x ^ { 2 } - 24 x = - ( x - 2 )$$

Solve for x

$$x=-\frac{1}{12}\approx -0.083333333$$

$$x=2$$

Steps Using Factoring By Grouping

To find the opposite of x-2, find the opposite of each term.

$$12x^{2}-24x=-x+2$$

Add x to both sides.

$$12x^{2}-24x+x=2$$

Combine -24x and x to get -23x.

$$12x^{2}-23x=2$$

Subtract 2 from both sides.

$$12x^{2}-23x-2=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 12x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=-23 ab=12\left(-2\right)=-24$$

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 -24.

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

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -23.

$$a=-24 b=1$$

Rewrite 12x^{2}-23x-2 as \left(12x^{2}-24x\right)+\left(x-2\right).

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

Factor out 12x in 12x^{2}-24x.

$$12x\left(x-2\right)+x-2$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(12x+1\right)$$

To find equation solutions, solve x-2=0 and 12x+1=0.

$$x=2 x=-\frac{1}{12}$$

Steps Using the Quadratic Formula

To find the opposite of x-2, find the opposite of each term.

$$12x^{2}-24x=-x+2$$

Add x to both sides.

$$12x^{2}-24x+x=2$$

Combine -24x and x to get -23x.

$$12x^{2}-23x=2$$

Subtract 2 from both sides.

$$12x^{2}-23x-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, -23 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\times 12\left(-2\right)}}{2\times 12}$$

Square -23.

$$x=\frac{-\left(-23\right)±\sqrt{529-4\times 12\left(-2\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-23\right)±\sqrt{529-48\left(-2\right)}}{2\times 12}$$

Multiply -48 times -2.

$$x=\frac{-\left(-23\right)±\sqrt{529+96}}{2\times 12}$$

Add 529 to 96.

$$x=\frac{-\left(-23\right)±\sqrt{625}}{2\times 12}$$

Take the square root of 625.

$$x=\frac{-\left(-23\right)±25}{2\times 12}$$

The opposite of -23 is 23.

$$x=\frac{23±25}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{23±25}{24}$$

Now solve the equation x=\frac{23±25}{24} when ± is plus. Add 23 to 25.

$$x=\frac{48}{24}$$

Divide 48 by 24.

$$x=2$$

Now solve the equation x=\frac{23±25}{24} when ± is minus. Subtract 25 from 23.

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

Reduce the fraction \frac{-2}{24} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{12}$$

The equation is now solved.

$$x=2 x=-\frac{1}{12}$$

Steps for Completing the Square

To find the opposite of x-2, find the opposite of each term.

$$12x^{2}-24x=-x+2$$

Add x to both sides.

$$12x^{2}-24x+x=2$$

Combine -24x and x to get -23x.

$$12x^{2}-23x=2$$

Divide both sides by 12.

$$\frac{12x^{2}-23x}{12}=\frac{2}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}-\frac{23}{12}x=\frac{2}{12}$$

Reduce the fraction \frac{2}{12} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{23}{12}x=\frac{1}{6}$$

Divide -\frac{23}{12}, the coefficient of the x term, by 2 to get -\frac{23}{24}. Then add the square of -\frac{23}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{23}{12}x+\left(-\frac{23}{24}\right)^{2}=\frac{1}{6}+\left(-\frac{23}{24}\right)^{2}$$

Square -\frac{23}{24} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{23}{12}x+\frac{529}{576}=\frac{1}{6}+\frac{529}{576}$$

Add \frac{1}{6} to \frac{529}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{23}{12}x+\frac{529}{576}=\frac{625}{576}$$

Factor x^{2}-\frac{23}{12}x+\frac{529}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{23}{24}\right)^{2}=\frac{625}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{23}{24}\right)^{2}}=\sqrt{\frac{625}{576}}$$

Simplify.

$$x-\frac{23}{24}=\frac{25}{24} x-\frac{23}{24}=-\frac{25}{24}$$

Add \frac{23}{24} to both sides of the equation.

$$x=2 x=-\frac{1}{12}$$