Solve 20/x+20+40/x=42/60

$$\frac { 20 } { x + 20 } + \frac { 40 } { x } = \frac { 42 } { 60 }$$

Solve for x

$$x = -\frac{100}{7} = -14\frac{2}{7} \approx -14.285714286$$

$$x=80$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -20,0 since division by zero is not defined. Multiply both sides of the equation by 60x\left(x+20\right), the least common multiple of x+20,x,60.

$$60x\times 20+\left(60x+1200\right)\times 40=x\left(x+20\right)\times 42$$

Multiply 60 and 20 to get 1200.

$$1200x+\left(60x+1200\right)\times 40=x\left(x+20\right)\times 42$$

Use the distributive property to multiply 60x+1200 by 40.

$$1200x+2400x+48000=x\left(x+20\right)\times 42$$

Combine 1200x and 2400x to get 3600x.

$$3600x+48000=x\left(x+20\right)\times 42$$

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

$$3600x+48000=\left(x^{2}+20x\right)\times 42$$

Use the distributive property to multiply x^{2}+20x by 42.

$$3600x+48000=42x^{2}+840x$$

Subtract 42x^{2} from both sides.

$$3600x+48000-42x^{2}=840x$$

Subtract 840x from both sides.

$$3600x+48000-42x^{2}-840x=0$$

Combine 3600x and -840x to get 2760x.

$$2760x+48000-42x^{2}=0$$

Divide both sides by 6.

$$460x+8000-7x^{2}=0$$

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

$$-7x^{2}+460x+8000=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -7x^{2}+ax+bx+8000. To find a and b, set up a system to be solved.

$$a+b=460 ab=-7\times 8000=-56000$$

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 -56000.

$$-1,56000 -2,28000 -4,14000 -5,11200 -7,8000 -8,7000 -10,5600 -14,4000 -16,3500 -20,2800 -25,2240 -28,2000 -32,1750 -35,1600 -40,1400 -50,1120 -56,1000 -64,875 -70,800 -80,700 -100,560 -112,500 -125,448 -140,400 -160,350 -175,320 -200,280 -224,250$$

Calculate the sum for each pair.

$$-1+56000=55999 -2+28000=27998 -4+14000=13996 -5+11200=11195 -7+8000=7993 -8+7000=6992 -10+5600=5590 -14+4000=3986 -16+3500=3484 -20+2800=2780 -25+2240=2215 -28+2000=1972 -32+1750=1718 -35+1600=1565 -40+1400=1360 -50+1120=1070 -56+1000=944 -64+875=811 -70+800=730 -80+700=620 -100+560=460 -112+500=388 -125+448=323 -140+400=260 -160+350=190 -175+320=145 -200+280=80 -224+250=26$$

The solution is the pair that gives sum 460.

$$a=560 b=-100$$

Rewrite -7x^{2}+460x+8000 as \left(-7x^{2}+560x\right)+\left(-100x+8000\right).

$$\left(-7x^{2}+560x\right)+\left(-100x+8000\right)$$

Factor out 7x in the first and 100 in the second group.

$$7x\left(-x+80\right)+100\left(-x+80\right)$$

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

$$\left(-x+80\right)\left(7x+100\right)$$

To find equation solutions, solve -x+80=0 and 7x+100=0.

$$x=80 x=-\frac{100}{7}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -20,0 since division by zero is not defined. Multiply both sides of the equation by 60x\left(x+20\right), the least common multiple of x+20,x,60.

$$60x\times 20+\left(60x+1200\right)\times 40=x\left(x+20\right)\times 42$$

Multiply 60 and 20 to get 1200.

$$1200x+\left(60x+1200\right)\times 40=x\left(x+20\right)\times 42$$

Use the distributive property to multiply 60x+1200 by 40.

$$1200x+2400x+48000=x\left(x+20\right)\times 42$$

Combine 1200x and 2400x to get 3600x.

$$3600x+48000=x\left(x+20\right)\times 42$$

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

$$3600x+48000=\left(x^{2}+20x\right)\times 42$$

Use the distributive property to multiply x^{2}+20x by 42.

$$3600x+48000=42x^{2}+840x$$

Subtract 42x^{2} from both sides.

$$3600x+48000-42x^{2}=840x$$

Subtract 840x from both sides.

$$3600x+48000-42x^{2}-840x=0$$

Combine 3600x and -840x to get 2760x.

$$2760x+48000-42x^{2}=0$$

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.

$$-42x^{2}+2760x+48000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -42 for a, 2760 for b, and 48000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2760±\sqrt{2760^{2}-4\left(-42\right)\times 48000}}{2\left(-42\right)}$$

Square 2760.

$$x=\frac{-2760±\sqrt{7617600-4\left(-42\right)\times 48000}}{2\left(-42\right)}$$

Multiply -4 times -42.

$$x=\frac{-2760±\sqrt{7617600+168\times 48000}}{2\left(-42\right)}$$

Multiply 168 times 48000.

$$x=\frac{-2760±\sqrt{7617600+8064000}}{2\left(-42\right)}$$

Add 7617600 to 8064000.

$$x=\frac{-2760±\sqrt{15681600}}{2\left(-42\right)}$$

Take the square root of 15681600.

$$x=\frac{-2760±3960}{2\left(-42\right)}$$

Multiply 2 times -42.

$$x=\frac{-2760±3960}{-84}$$

Now solve the equation x=\frac{-2760±3960}{-84} when ± is plus. Add -2760 to 3960.

$$x=\frac{1200}{-84}$$

Reduce the fraction \frac{1200}{-84} to lowest terms by extracting and canceling out 12.

$$x=-\frac{100}{7}$$

Now solve the equation x=\frac{-2760±3960}{-84} when ± is minus. Subtract 3960 from -2760.

$$x=-\frac{6720}{-84}$$

Divide -6720 by -84.

$$x=80$$

The equation is now solved.

$$x=-\frac{100}{7} x=80$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -20,0 since division by zero is not defined. Multiply both sides of the equation by 60x\left(x+20\right), the least common multiple of x+20,x,60.

$$60x\times 20+\left(60x+1200\right)\times 40=x\left(x+20\right)\times 42$$

Multiply 60 and 20 to get 1200.

$$1200x+\left(60x+1200\right)\times 40=x\left(x+20\right)\times 42$$

Use the distributive property to multiply 60x+1200 by 40.

$$1200x+2400x+48000=x\left(x+20\right)\times 42$$

Combine 1200x and 2400x to get 3600x.

$$3600x+48000=x\left(x+20\right)\times 42$$

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

$$3600x+48000=\left(x^{2}+20x\right)\times 42$$

Use the distributive property to multiply x^{2}+20x by 42.

$$3600x+48000=42x^{2}+840x$$

Subtract 42x^{2} from both sides.

$$3600x+48000-42x^{2}=840x$$

Subtract 840x from both sides.

$$3600x+48000-42x^{2}-840x=0$$

Combine 3600x and -840x to get 2760x.

$$2760x+48000-42x^{2}=0$$

Subtract 48000 from both sides. Anything subtracted from zero gives its negation.

$$2760x-42x^{2}=-48000$$

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.

$$-42x^{2}+2760x=-48000$$

Divide both sides by -42.

$$\frac{-42x^{2}+2760x}{-42}=-\frac{48000}{-42}$$

Dividing by -42 undoes the multiplication by -42.

$$x^{2}+\frac{2760}{-42}x=-\frac{48000}{-42}$$

Reduce the fraction \frac{2760}{-42} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{460}{7}x=-\frac{48000}{-42}$$

Reduce the fraction \frac{-48000}{-42} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{460}{7}x=\frac{8000}{7}$$

Divide -\frac{460}{7}, the coefficient of the x term, by 2 to get -\frac{230}{7}. Then add the square of -\frac{230}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{460}{7}x+\left(-\frac{230}{7}\right)^{2}=\frac{8000}{7}+\left(-\frac{230}{7}\right)^{2}$$

Square -\frac{230}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{460}{7}x+\frac{52900}{49}=\frac{8000}{7}+\frac{52900}{49}$$

Add \frac{8000}{7} to \frac{52900}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{460}{7}x+\frac{52900}{49}=\frac{108900}{49}$$

Factor x^{2}-\frac{460}{7}x+\frac{52900}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{230}{7}\right)^{2}=\frac{108900}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{230}{7}\right)^{2}}=\sqrt{\frac{108900}{49}}$$

Simplify.

$$x-\frac{230}{7}=\frac{330}{7} x-\frac{230}{7}=-\frac{330}{7}$$

Add \frac{230}{7} to both sides of the equation.

$$x=80 x=-\frac{100}{7}$$