Solve 96/x-96/x+12=2/3

$$\frac{ 96 }{ x } - \frac{ 96 }{ x+12 } = \frac{ 2 }{ 3 }$$

Solve for x

$$x=-48$$

$$x=36$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -12,0 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x+12\right), the least common multiple of x,x+12,3.

$$\left(3x+36\right)\times 96-3x\times 96=2x\left(x+12\right)$$

Use the distributive property to multiply 3x+36 by 96.

$$288x+3456-3x\times 96=2x\left(x+12\right)$$

Multiply 3 and 96 to get 288.

$$288x+3456-288x=2x\left(x+12\right)$$

Use the distributive property to multiply 2x by x+12.

$$288x+3456-288x=2x^{2}+24x$$

Subtract 2x^{2} from both sides.

$$288x+3456-288x-2x^{2}=24x$$

Subtract 24x from both sides.

$$288x+3456-288x-2x^{2}-24x=0$$

Combine 288x and -24x to get 264x.

$$264x+3456-288x-2x^{2}=0$$

Combine 264x and -288x to get -24x.

$$-24x+3456-2x^{2}=0$$

Divide both sides by 2.

$$-12x+1728-x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}-12x+1728=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+1728. To find a and b, set up a system to be solved.

$$a+b=-12 ab=-1728=-1728$$

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 -1728.

$$1,-1728 2,-864 3,-576 4,-432 6,-288 8,-216 9,-192 12,-144 16,-108 18,-96 24,-72 27,-64 32,-54 36,-48$$

Calculate the sum for each pair.

$$1-1728=-1727 2-864=-862 3-576=-573 4-432=-428 6-288=-282 8-216=-208 9-192=-183 12-144=-132 16-108=-92 18-96=-78 24-72=-48 27-64=-37 32-54=-22 36-48=-12$$

The solution is the pair that gives sum -12.

$$a=36 b=-48$$

Rewrite -x^{2}-12x+1728 as \left(-x^{2}+36x\right)+\left(-48x+1728\right).

$$\left(-x^{2}+36x\right)+\left(-48x+1728\right)$$

Factor out x in the first and 48 in the second group.

$$x\left(-x+36\right)+48\left(-x+36\right)$$

Factor out common term -x+36 by using distributive property.

$$\left(-x+36\right)\left(x+48\right)$$

To find equation solutions, solve -x+36=0 and x+48=0.

$$x=36 x=-48$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -12,0 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x+12\right), the least common multiple of x,x+12,3.

$$\left(3x+36\right)\times 96-3x\times 96=2x\left(x+12\right)$$

Use the distributive property to multiply 3x+36 by 96.

$$288x+3456-3x\times 96=2x\left(x+12\right)$$

Multiply 3 and 96 to get 288.

$$288x+3456-288x=2x\left(x+12\right)$$

Use the distributive property to multiply 2x by x+12.

$$288x+3456-288x=2x^{2}+24x$$

Subtract 2x^{2} from both sides.

$$288x+3456-288x-2x^{2}=24x$$

Subtract 24x from both sides.

$$288x+3456-288x-2x^{2}-24x=0$$

Combine 288x and -24x to get 264x.

$$264x+3456-288x-2x^{2}=0$$

Combine 264x and -288x to get -24x.

$$-24x+3456-2x^{2}=0$$

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.

$$-2x^{2}-24x+3456=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -24 for b, and 3456 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\left(-2\right)\times 3456}}{2\left(-2\right)}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\left(-2\right)\times 3456}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-24\right)±\sqrt{576+8\times 3456}}{2\left(-2\right)}$$

Multiply 8 times 3456.

$$x=\frac{-\left(-24\right)±\sqrt{576+27648}}{2\left(-2\right)}$$

Add 576 to 27648.

$$x=\frac{-\left(-24\right)±\sqrt{28224}}{2\left(-2\right)}$$

Take the square root of 28224.

$$x=\frac{-\left(-24\right)±168}{2\left(-2\right)}$$

The opposite of -24 is 24.

$$x=\frac{24±168}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{24±168}{-4}$$

Now solve the equation x=\frac{24±168}{-4} when ± is plus. Add 24 to 168.

$$x=\frac{192}{-4}$$

Divide 192 by -4.

$$x=-48$$

Now solve the equation x=\frac{24±168}{-4} when ± is minus. Subtract 168 from 24.

$$x=-\frac{144}{-4}$$

Divide -144 by -4.

$$x=36$$

The equation is now solved.

$$x=-48 x=36$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -12,0 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x+12\right), the least common multiple of x,x+12,3.

$$\left(3x+36\right)\times 96-3x\times 96=2x\left(x+12\right)$$

Use the distributive property to multiply 3x+36 by 96.

$$288x+3456-3x\times 96=2x\left(x+12\right)$$

Multiply 3 and 96 to get 288.

$$288x+3456-288x=2x\left(x+12\right)$$

Use the distributive property to multiply 2x by x+12.

$$288x+3456-288x=2x^{2}+24x$$

Subtract 2x^{2} from both sides.

$$288x+3456-288x-2x^{2}=24x$$

Subtract 24x from both sides.

$$288x+3456-288x-2x^{2}-24x=0$$

Combine 288x and -24x to get 264x.

$$264x+3456-288x-2x^{2}=0$$

Subtract 3456 from both sides. Anything subtracted from zero gives its negation.

$$264x-288x-2x^{2}=-3456$$

Combine 264x and -288x to get -24x.

$$-24x-2x^{2}=-3456$$

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.

$$-2x^{2}-24x=-3456$$

Divide both sides by -2.

$$\frac{-2x^{2}-24x}{-2}=-\frac{3456}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{24}{-2}\right)x=-\frac{3456}{-2}$$

Divide -24 by -2.

$$x^{2}+12x=-\frac{3456}{-2}$$

Divide -3456 by -2.

$$x^{2}+12x=1728$$

Divide 12, the coefficient of the x term, by 2 to get 6. Then add the square of 6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+12x+6^{2}=1728+6^{2}$$

Square 6.

$$x^{2}+12x+36=1728+36$$

Add 1728 to 36.

$$x^{2}+12x+36=1764$$

Factor x^{2}+12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+6\right)^{2}=1764$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{1764}$$

Simplify.

$$x+6=42 x+6=-42$$

Subtract 6 from both sides of the equation.

$$x=36 x=-48$$