Solve 4800/x+10=4200/x-20

$$\frac{ 4800 }{ x } +10= \frac{ 4200 }{ x-20 }$$

Solve for x

$$x=-120$$

$$x=80$$

Steps Using Factoring By Grouping

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

$$\left(x-20\right)\times 4800+x\left(x-20\right)\times 10=x\times 4200$$

Use the distributive property to multiply x-20 by 4800.

$$4800x-96000+x\left(x-20\right)\times 10=x\times 4200$$

Use the distributive property to multiply x by x-20.

$$4800x-96000+\left(x^{2}-20x\right)\times 10=x\times 4200$$

Use the distributive property to multiply x^{2}-20x by 10.

$$4800x-96000+10x^{2}-200x=x\times 4200$$

Combine 4800x and -200x to get 4600x.

$$4600x-96000+10x^{2}=x\times 4200$$

Subtract x\times 4200 from both sides.

$$4600x-96000+10x^{2}-x\times 4200=0$$

Combine 4600x and -x\times 4200 to get 400x.

$$400x-96000+10x^{2}=0$$

Divide both sides by 10.

$$40x-9600+x^{2}=0$$

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

$$x^{2}+40x-9600=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-9600. To find a and b, set up a system to be solved.

$$a+b=40 ab=1\left(-9600\right)=-9600$$

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 -9600.

$$-1,9600 -2,4800 -3,3200 -4,2400 -5,1920 -6,1600 -8,1200 -10,960 -12,800 -15,640 -16,600 -20,480 -24,400 -25,384 -30,320 -32,300 -40,240 -48,200 -50,192 -60,160 -64,150 -75,128 -80,120 -96,100$$

Calculate the sum for each pair.

$$-1+9600=9599 -2+4800=4798 -3+3200=3197 -4+2400=2396 -5+1920=1915 -6+1600=1594 -8+1200=1192 -10+960=950 -12+800=788 -15+640=625 -16+600=584 -20+480=460 -24+400=376 -25+384=359 -30+320=290 -32+300=268 -40+240=200 -48+200=152 -50+192=142 -60+160=100 -64+150=86 -75+128=53 -80+120=40 -96+100=4$$

The solution is the pair that gives sum 40.

$$a=-80 b=120$$

Rewrite x^{2}+40x-9600 as \left(x^{2}-80x\right)+\left(120x-9600\right).

$$\left(x^{2}-80x\right)+\left(120x-9600\right)$$

Factor out x in the first and 120 in the second group.

$$x\left(x-80\right)+120\left(x-80\right)$$

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

$$\left(x-80\right)\left(x+120\right)$$

To find equation solutions, solve x-80=0 and x+120=0.

$$x=80 x=-120$$

Steps Using the Quadratic Formula

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

$$\left(x-20\right)\times 4800+x\left(x-20\right)\times 10=x\times 4200$$

Use the distributive property to multiply x-20 by 4800.

$$4800x-96000+x\left(x-20\right)\times 10=x\times 4200$$

Use the distributive property to multiply x by x-20.

$$4800x-96000+\left(x^{2}-20x\right)\times 10=x\times 4200$$

Use the distributive property to multiply x^{2}-20x by 10.

$$4800x-96000+10x^{2}-200x=x\times 4200$$

Combine 4800x and -200x to get 4600x.

$$4600x-96000+10x^{2}=x\times 4200$$

Subtract x\times 4200 from both sides.

$$4600x-96000+10x^{2}-x\times 4200=0$$

Combine 4600x and -x\times 4200 to get 400x.

$$400x-96000+10x^{2}=0$$

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.

$$10x^{2}+400x-96000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, 400 for b, and -96000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-400±\sqrt{400^{2}-4\times 10\left(-96000\right)}}{2\times 10}$$

Square 400.

$$x=\frac{-400±\sqrt{160000-4\times 10\left(-96000\right)}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-400±\sqrt{160000-40\left(-96000\right)}}{2\times 10}$$

Multiply -40 times -96000.

$$x=\frac{-400±\sqrt{160000+3840000}}{2\times 10}$$

Add 160000 to 3840000.

$$x=\frac{-400±\sqrt{4000000}}{2\times 10}$$

Take the square root of 4000000.

$$x=\frac{-400±2000}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{-400±2000}{20}$$

Now solve the equation x=\frac{-400±2000}{20} when ± is plus. Add -400 to 2000.

$$x=\frac{1600}{20}$$

Divide 1600 by 20.

$$x=80$$

Now solve the equation x=\frac{-400±2000}{20} when ± is minus. Subtract 2000 from -400.

$$x=-\frac{2400}{20}$$

Divide -2400 by 20.

$$x=-120$$

The equation is now solved.

$$x=80 x=-120$$

Steps for Completing the Square

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

$$\left(x-20\right)\times 4800+x\left(x-20\right)\times 10=x\times 4200$$

Use the distributive property to multiply x-20 by 4800.

$$4800x-96000+x\left(x-20\right)\times 10=x\times 4200$$

Use the distributive property to multiply x by x-20.

$$4800x-96000+\left(x^{2}-20x\right)\times 10=x\times 4200$$

Use the distributive property to multiply x^{2}-20x by 10.

$$4800x-96000+10x^{2}-200x=x\times 4200$$

Combine 4800x and -200x to get 4600x.

$$4600x-96000+10x^{2}=x\times 4200$$

Subtract x\times 4200 from both sides.

$$4600x-96000+10x^{2}-x\times 4200=0$$

Combine 4600x and -x\times 4200 to get 400x.

$$400x-96000+10x^{2}=0$$

Add 96000 to both sides. Anything plus zero gives itself.

$$400x+10x^{2}=96000$$

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.

$$10x^{2}+400x=96000$$

Divide both sides by 10.

$$\frac{10x^{2}+400x}{10}=\frac{96000}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\frac{400}{10}x=\frac{96000}{10}$$

Divide 400 by 10.

$$x^{2}+40x=\frac{96000}{10}$$

Divide 96000 by 10.

$$x^{2}+40x=9600$$

Divide 40, the coefficient of the x term, by 2 to get 20. Then add the square of 20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+40x+20^{2}=9600+20^{2}$$

Square 20.

$$x^{2}+40x+400=9600+400$$

Add 9600 to 400.

$$x^{2}+40x+400=10000$$

Factor x^{2}+40x+400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+20\right)^{2}=10000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+20\right)^{2}}=\sqrt{10000}$$

Simplify.

$$x+20=100 x+20=-100$$

Subtract 20 from both sides of the equation.

$$x=80 x=-120$$