Solve 9/20-60/x+60/x+120=0

$$\frac { 9 } { 20 } - \frac { 60 } { x } + \frac { 60 } { x + 120 } = 0$$

Solve for x

$$x=-200$$

$$x=80$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -120,0 since division by zero is not defined. Multiply both sides of the equation by 20x\left(x+120\right), the least common multiple of 20,x,x+120.

$$20x\left(x+120\right)\times \frac{9}{20}-\left(20x+2400\right)\times 60+20x\times 60=0$$

Use the distributive property to multiply 20x by x+120.

$$\left(20x^{2}+2400x\right)\times \frac{9}{20}-\left(20x+2400\right)\times 60+20x\times 60=0$$

Use the distributive property to multiply 20x^{2}+2400x by \frac{9}{20}.

$$9x^{2}+1080x-\left(20x+2400\right)\times 60+20x\times 60=0$$

Use the distributive property to multiply 20x+2400 by 60.

$$9x^{2}+1080x-\left(1200x+144000\right)+20x\times 60=0$$

To find the opposite of 1200x+144000, find the opposite of each term.

$$9x^{2}+1080x-1200x-144000+20x\times 60=0$$

Combine 1080x and -1200x to get -120x.

$$9x^{2}-120x-144000+20x\times 60=0$$

Multiply 20 and 60 to get 1200.

$$9x^{2}-120x-144000+1200x=0$$

Combine -120x and 1200x to get 1080x.

$$9x^{2}+1080x-144000=0$$

Divide both sides by 9.

$$x^{2}+120x-16000=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-16000. To find a and b, set up a system to be solved.

$$a+b=120 ab=1\left(-16000\right)=-16000$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -16000.

$$-1,16000 -2,8000 -4,4000 -5,3200 -8,2000 -10,1600 -16,1000 -20,800 -25,640 -32,500 -40,400 -50,320 -64,250 -80,200 -100,160 -125,128$$

Calculate the sum for each pair.

$$-1+16000=15999 -2+8000=7998 -4+4000=3996 -5+3200=3195 -8+2000=1992 -10+1600=1590 -16+1000=984 -20+800=780 -25+640=615 -32+500=468 -40+400=360 -50+320=270 -64+250=186 -80+200=120 -100+160=60 -125+128=3$$

The solution is the pair that gives sum 120.

$$a=-80 b=200$$

Rewrite x^{2}+120x-16000 as \left(x^{2}-80x\right)+\left(200x-16000\right).

$$\left(x^{2}-80x\right)+\left(200x-16000\right)$$

Factor out x in the first and 200 in the second group.

$$x\left(x-80\right)+200\left(x-80\right)$$

Factor out common term x-80 by using distributive property.

$$\left(x-80\right)\left(x+200\right)$$

To find equation solutions, solve x-80=0 and x+200=0.

$$x=80 x=-200$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -120,0 since division by zero is not defined. Multiply both sides of the equation by 20x\left(x+120\right), the least common multiple of 20,x,x+120.

$$20x\left(x+120\right)\times \frac{9}{20}-\left(20x+2400\right)\times 60+20x\times 60=0$$

Use the distributive property to multiply 20x by x+120.

$$\left(20x^{2}+2400x\right)\times \frac{9}{20}-\left(20x+2400\right)\times 60+20x\times 60=0$$

Use the distributive property to multiply 20x^{2}+2400x by \frac{9}{20}.

$$9x^{2}+1080x-\left(20x+2400\right)\times 60+20x\times 60=0$$

Use the distributive property to multiply 20x+2400 by 60.

$$9x^{2}+1080x-\left(1200x+144000\right)+20x\times 60=0$$

To find the opposite of 1200x+144000, find the opposite of each term.

$$9x^{2}+1080x-1200x-144000+20x\times 60=0$$

Combine 1080x and -1200x to get -120x.

$$9x^{2}-120x-144000+20x\times 60=0$$

Multiply 20 and 60 to get 1200.

$$9x^{2}-120x-144000+1200x=0$$

Combine -120x and 1200x to get 1080x.

$$9x^{2}+1080x-144000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 1080 for b, and -144000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1080±\sqrt{1080^{2}-4\times 9\left(-144000\right)}}{2\times 9}$$

Square 1080.

$$x=\frac{-1080±\sqrt{1166400-4\times 9\left(-144000\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-1080±\sqrt{1166400-36\left(-144000\right)}}{2\times 9}$$

Multiply -36 times -144000.

$$x=\frac{-1080±\sqrt{1166400+5184000}}{2\times 9}$$

Add 1166400 to 5184000.

$$x=\frac{-1080±\sqrt{6350400}}{2\times 9}$$

Take the square root of 6350400.

$$x=\frac{-1080±2520}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-1080±2520}{18}$$

Now solve the equation x=\frac{-1080±2520}{18} when ± is plus. Add -1080 to 2520.

$$x=\frac{1440}{18}$$

Divide 1440 by 18.

$$x=80$$

Now solve the equation x=\frac{-1080±2520}{18} when ± is minus. Subtract 2520 from -1080.

$$x=-\frac{3600}{18}$$

Divide -3600 by 18.

$$x=-200$$

The equation is now solved.

$$x=80 x=-200$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -120,0 since division by zero is not defined. Multiply both sides of the equation by 20x\left(x+120\right), the least common multiple of 20,x,x+120.

$$20x\left(x+120\right)\times \frac{9}{20}-\left(20x+2400\right)\times 60+20x\times 60=0$$

Use the distributive property to multiply 20x by x+120.

$$\left(20x^{2}+2400x\right)\times \frac{9}{20}-\left(20x+2400\right)\times 60+20x\times 60=0$$

Use the distributive property to multiply 20x^{2}+2400x by \frac{9}{20}.

$$9x^{2}+1080x-\left(20x+2400\right)\times 60+20x\times 60=0$$

Use the distributive property to multiply 20x+2400 by 60.

$$9x^{2}+1080x-\left(1200x+144000\right)+20x\times 60=0$$

To find the opposite of 1200x+144000, find the opposite of each term.

$$9x^{2}+1080x-1200x-144000+20x\times 60=0$$

Combine 1080x and -1200x to get -120x.

$$9x^{2}-120x-144000+20x\times 60=0$$

Multiply 20 and 60 to get 1200.

$$9x^{2}-120x-144000+1200x=0$$

Combine -120x and 1200x to get 1080x.

$$9x^{2}+1080x-144000=0$$

Add 144000 to both sides. Anything plus zero gives itself.

$$9x^{2}+1080x=144000$$

Divide both sides by 9.

$$\frac{9x^{2}+1080x}{9}=\frac{144000}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\frac{1080}{9}x=\frac{144000}{9}$$

Divide 1080 by 9.

$$x^{2}+120x=\frac{144000}{9}$$

Divide 144000 by 9.

$$x^{2}+120x=16000$$

Divide 120, the coefficient of the x term, by 2 to get 60. Then add the square of 60 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+120x+60^{2}=16000+60^{2}$$

Square 60.

$$x^{2}+120x+3600=16000+3600$$

Add 16000 to 3600.

$$x^{2}+120x+3600=19600$$

Factor x^{2}+120x+3600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+60\right)^{2}=19600$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+60\right)^{2}}=\sqrt{19600}$$

Simplify.

$$x+60=140 x+60=-140$$

Subtract 60 from both sides of the equation.

$$x=80 x=-200$$