Solve 180/x-180/x+2=3

$$\frac { 180 } { x } - \frac { 180 } { x + 2 } = 3$$

Solve for x

$$x=-12$$

$$x=10$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+2\right), the least common multiple of x,x+2.

$$\left(x+2\right)\times 180-x\times 180=3x\left(x+2\right)$$

Use the distributive property to multiply x+2 by 180.

$$180x+360-x\times 180=3x\left(x+2\right)$$

Use the distributive property to multiply 3x by x+2.

$$180x+360-x\times 180=3x^{2}+6x$$

Subtract 3x^{2} from both sides.

$$180x+360-x\times 180-3x^{2}=6x$$

Subtract 6x from both sides.

$$180x+360-x\times 180-3x^{2}-6x=0$$

Combine 180x and -6x to get 174x.

$$174x+360-x\times 180-3x^{2}=0$$

Multiply -1 and 180 to get -180.

$$174x+360-180x-3x^{2}=0$$

Combine 174x and -180x to get -6x.

$$-6x+360-3x^{2}=0$$

Divide both sides by 3.

$$-2x+120-x^{2}=0$$

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

$$-x^{2}-2x+120=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+120. To find a and b, set up a system to be solved.

$$a+b=-2 ab=-120=-120$$

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 -120.

$$1,-120 2,-60 3,-40 4,-30 5,-24 6,-20 8,-15 10,-12$$

Calculate the sum for each pair.

$$1-120=-119 2-60=-58 3-40=-37 4-30=-26 5-24=-19 6-20=-14 8-15=-7 10-12=-2$$

The solution is the pair that gives sum -2.

$$a=10 b=-12$$

Rewrite -x^{2}-2x+120 as \left(-x^{2}+10x\right)+\left(-12x+120\right).

$$\left(-x^{2}+10x\right)+\left(-12x+120\right)$$

Factor out x in the first and 12 in the second group.

$$x\left(-x+10\right)+12\left(-x+10\right)$$

Factor out common term -x+10 by using distributive property.

$$\left(-x+10\right)\left(x+12\right)$$

To find equation solutions, solve -x+10=0 and x+12=0.

$$x=10 x=-12$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+2\right), the least common multiple of x,x+2.

$$\left(x+2\right)\times 180-x\times 180=3x\left(x+2\right)$$

Use the distributive property to multiply x+2 by 180.

$$180x+360-x\times 180=3x\left(x+2\right)$$

Use the distributive property to multiply 3x by x+2.

$$180x+360-x\times 180=3x^{2}+6x$$

Subtract 3x^{2} from both sides.

$$180x+360-x\times 180-3x^{2}=6x$$

Subtract 6x from both sides.

$$180x+360-x\times 180-3x^{2}-6x=0$$

Combine 180x and -6x to get 174x.

$$174x+360-x\times 180-3x^{2}=0$$

Multiply -1 and 180 to get -180.

$$174x+360-180x-3x^{2}=0$$

Combine 174x and -180x to get -6x.

$$-6x+360-3x^{2}=0$$

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.

$$-3x^{2}-6x+360=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -6 for b, and 360 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-3\right)\times 360}}{2\left(-3\right)}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-3\right)\times 360}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-6\right)±\sqrt{36+12\times 360}}{2\left(-3\right)}$$

Multiply 12 times 360.

$$x=\frac{-\left(-6\right)±\sqrt{36+4320}}{2\left(-3\right)}$$

Add 36 to 4320.

$$x=\frac{-\left(-6\right)±\sqrt{4356}}{2\left(-3\right)}$$

Take the square root of 4356.

$$x=\frac{-\left(-6\right)±66}{2\left(-3\right)}$$

The opposite of -6 is 6.

$$x=\frac{6±66}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{6±66}{-6}$$

Now solve the equation x=\frac{6±66}{-6} when ± is plus. Add 6 to 66.

$$x=\frac{72}{-6}$$

Divide 72 by -6.

$$x=-12$$

Now solve the equation x=\frac{6±66}{-6} when ± is minus. Subtract 66 from 6.

$$x=-\frac{60}{-6}$$

Divide -60 by -6.

$$x=10$$

The equation is now solved.

$$x=-12 x=10$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+2\right), the least common multiple of x,x+2.

$$\left(x+2\right)\times 180-x\times 180=3x\left(x+2\right)$$

Use the distributive property to multiply x+2 by 180.

$$180x+360-x\times 180=3x\left(x+2\right)$$

Use the distributive property to multiply 3x by x+2.

$$180x+360-x\times 180=3x^{2}+6x$$

Subtract 3x^{2} from both sides.

$$180x+360-x\times 180-3x^{2}=6x$$

Subtract 6x from both sides.

$$180x+360-x\times 180-3x^{2}-6x=0$$

Combine 180x and -6x to get 174x.

$$174x+360-x\times 180-3x^{2}=0$$

Subtract 360 from both sides. Anything subtracted from zero gives its negation.

$$174x-x\times 180-3x^{2}=-360$$

Multiply -1 and 180 to get -180.

$$174x-180x-3x^{2}=-360$$

Combine 174x and -180x to get -6x.

$$-6x-3x^{2}=-360$$

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.

$$-3x^{2}-6x=-360$$

Divide both sides by -3.

$$\frac{-3x^{2}-6x}{-3}=-\frac{360}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{6}{-3}\right)x=-\frac{360}{-3}$$

Divide -6 by -3.

$$x^{2}+2x=-\frac{360}{-3}$$

Divide -360 by -3.

$$x^{2}+2x=120$$

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.

$$x^{2}+2x+1^{2}=120+1^{2}$$

Square 1.

$$x^{2}+2x+1=120+1$$

Add 120 to 1.

$$x^{2}+2x+1=121$$

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}.

$$\left(x+1\right)^{2}=121$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{121}$$

Simplify.

$$x+1=11 x+1=-11$$

Subtract 1 from both sides of the equation.

$$x=10 x=-12$$