Solve 400/x+20+400/52/x+20+frac{400/53}{x}=11

$$\frac{ 400 }{ x+20 } + \frac{ \frac{ 400 }{ 5 } 2 }{ x+20 } + \frac{ \frac{ 400 }{ 5 } 3 }{ x } = 11$$

Solve for x

$$x = -\frac{80}{11} = -7\frac{3}{11} \approx -7.272727273$$

$$x=60$$

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 x\left(x+20\right), the least common multiple of x+20,x.

$$x\times 400+x\times \frac{400}{5}\times 2+\left(x+20\right)\times \frac{400}{5}\times 3=11x\left(x+20\right)$$

Divide 400 by 5 to get 80.

$$x\times 400+x\times 80\times 2+\left(x+20\right)\times \frac{400}{5}\times 3=11x\left(x+20\right)$$

Multiply 80 and 2 to get 160.

$$x\times 400+x\times 160+\left(x+20\right)\times \frac{400}{5}\times 3=11x\left(x+20\right)$$

Combine x\times 400 and x\times 160 to get 560x.

$$560x+\left(x+20\right)\times \frac{400}{5}\times 3=11x\left(x+20\right)$$

Divide 400 by 5 to get 80.

$$560x+\left(x+20\right)\times 80\times 3=11x\left(x+20\right)$$

Multiply 80 and 3 to get 240.

$$560x+\left(x+20\right)\times 240=11x\left(x+20\right)$$

Use the distributive property to multiply x+20 by 240.

$$560x+240x+4800=11x\left(x+20\right)$$

Combine 560x and 240x to get 800x.

$$800x+4800=11x\left(x+20\right)$$

Use the distributive property to multiply 11x by x+20.

$$800x+4800=11x^{2}+220x$$

Subtract 11x^{2} from both sides.

$$800x+4800-11x^{2}=220x$$

Subtract 220x from both sides.

$$800x+4800-11x^{2}-220x=0$$

Combine 800x and -220x to get 580x.

$$580x+4800-11x^{2}=0$$

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

$$-11x^{2}+580x+4800=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -11x^{2}+ax+bx+4800. To find a and b, set up a system to be solved.

$$a+b=580 ab=-11\times 4800=-52800$$

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 -52800.

$$-1,52800 -2,26400 -3,17600 -4,13200 -5,10560 -6,8800 -8,6600 -10,5280 -11,4800 -12,4400 -15,3520 -16,3300 -20,2640 -22,2400 -24,2200 -25,2112 -30,1760 -32,1650 -33,1600 -40,1320 -44,1200 -48,1100 -50,1056 -55,960 -60,880 -64,825 -66,800 -75,704 -80,660 -88,600 -96,550 -100,528 -110,480 -120,440 -132,400 -150,352 -160,330 -165,320 -176,300 -192,275 -200,264 -220,240$$

Calculate the sum for each pair.

$$-1+52800=52799 -2+26400=26398 -3+17600=17597 -4+13200=13196 -5+10560=10555 -6+8800=8794 -8+6600=6592 -10+5280=5270 -11+4800=4789 -12+4400=4388 -15+3520=3505 -16+3300=3284 -20+2640=2620 -22+2400=2378 -24+2200=2176 -25+2112=2087 -30+1760=1730 -32+1650=1618 -33+1600=1567 -40+1320=1280 -44+1200=1156 -48+1100=1052 -50+1056=1006 -55+960=905 -60+880=820 -64+825=761 -66+800=734 -75+704=629 -80+660=580 -88+600=512 -96+550=454 -100+528=428 -110+480=370 -120+440=320 -132+400=268 -150+352=202 -160+330=170 -165+320=155 -176+300=124 -192+275=83 -200+264=64 -220+240=20$$

The solution is the pair that gives sum 580.

$$a=660 b=-80$$

Rewrite -11x^{2}+580x+4800 as \left(-11x^{2}+660x\right)+\left(-80x+4800\right).

$$\left(-11x^{2}+660x\right)+\left(-80x+4800\right)$$

Factor out 11x in the first and 80 in the second group.

$$11x\left(-x+60\right)+80\left(-x+60\right)$$

Factor out common term -x+60 by using distributive property.

$$\left(-x+60\right)\left(11x+80\right)$$

To find equation solutions, solve -x+60=0 and 11x+80=0.

$$x=60 x=-\frac{80}{11}$$

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 x\left(x+20\right), the least common multiple of x+20,x.

$$x\times 400+x\times \frac{400}{5}\times 2+\left(x+20\right)\times \frac{400}{5}\times 3=11x\left(x+20\right)$$

Divide 400 by 5 to get 80.

$$x\times 400+x\times 80\times 2+\left(x+20\right)\times \frac{400}{5}\times 3=11x\left(x+20\right)$$

Multiply 80 and 2 to get 160.

$$x\times 400+x\times 160+\left(x+20\right)\times \frac{400}{5}\times 3=11x\left(x+20\right)$$

Combine x\times 400 and x\times 160 to get 560x.

$$560x+\left(x+20\right)\times \frac{400}{5}\times 3=11x\left(x+20\right)$$

Divide 400 by 5 to get 80.

$$560x+\left(x+20\right)\times 80\times 3=11x\left(x+20\right)$$

Multiply 80 and 3 to get 240.

$$560x+\left(x+20\right)\times 240=11x\left(x+20\right)$$

Use the distributive property to multiply x+20 by 240.

$$560x+240x+4800=11x\left(x+20\right)$$

Combine 560x and 240x to get 800x.

$$800x+4800=11x\left(x+20\right)$$

Use the distributive property to multiply 11x by x+20.

$$800x+4800=11x^{2}+220x$$

Subtract 11x^{2} from both sides.

$$800x+4800-11x^{2}=220x$$

Subtract 220x from both sides.

$$800x+4800-11x^{2}-220x=0$$

Combine 800x and -220x to get 580x.

$$580x+4800-11x^{2}=0$$

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.

$$-11x^{2}+580x+4800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -11 for a, 580 for b, and 4800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-580±\sqrt{580^{2}-4\left(-11\right)\times 4800}}{2\left(-11\right)}$$

Square 580.

$$x=\frac{-580±\sqrt{336400-4\left(-11\right)\times 4800}}{2\left(-11\right)}$$

Multiply -4 times -11.

$$x=\frac{-580±\sqrt{336400+44\times 4800}}{2\left(-11\right)}$$

Multiply 44 times 4800.

$$x=\frac{-580±\sqrt{336400+211200}}{2\left(-11\right)}$$

Add 336400 to 211200.

$$x=\frac{-580±\sqrt{547600}}{2\left(-11\right)}$$

Take the square root of 547600.

$$x=\frac{-580±740}{2\left(-11\right)}$$

Multiply 2 times -11.

$$x=\frac{-580±740}{-22}$$

Now solve the equation x=\frac{-580±740}{-22} when ± is plus. Add -580 to 740.

$$x=\frac{160}{-22}$$

Reduce the fraction \frac{160}{-22} to lowest terms by extracting and canceling out 2.

$$x=-\frac{80}{11}$$

Now solve the equation x=\frac{-580±740}{-22} when ± is minus. Subtract 740 from -580.

$$x=-\frac{1320}{-22}$$

Divide -1320 by -22.

$$x=60$$

The equation is now solved.

$$x=-\frac{80}{11} x=60$$

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 x\left(x+20\right), the least common multiple of x+20,x.

$$x\times 400+x\times \frac{400}{5}\times 2+\left(x+20\right)\times \frac{400}{5}\times 3=11x\left(x+20\right)$$

Divide 400 by 5 to get 80.

$$x\times 400+x\times 80\times 2+\left(x+20\right)\times \frac{400}{5}\times 3=11x\left(x+20\right)$$

Multiply 80 and 2 to get 160.

$$x\times 400+x\times 160+\left(x+20\right)\times \frac{400}{5}\times 3=11x\left(x+20\right)$$

Combine x\times 400 and x\times 160 to get 560x.

$$560x+\left(x+20\right)\times \frac{400}{5}\times 3=11x\left(x+20\right)$$

Divide 400 by 5 to get 80.

$$560x+\left(x+20\right)\times 80\times 3=11x\left(x+20\right)$$

Multiply 80 and 3 to get 240.

$$560x+\left(x+20\right)\times 240=11x\left(x+20\right)$$

Use the distributive property to multiply x+20 by 240.

$$560x+240x+4800=11x\left(x+20\right)$$

Combine 560x and 240x to get 800x.

$$800x+4800=11x\left(x+20\right)$$

Use the distributive property to multiply 11x by x+20.

$$800x+4800=11x^{2}+220x$$

Subtract 11x^{2} from both sides.

$$800x+4800-11x^{2}=220x$$

Subtract 220x from both sides.

$$800x+4800-11x^{2}-220x=0$$

Combine 800x and -220x to get 580x.

$$580x+4800-11x^{2}=0$$

Subtract 4800 from both sides. Anything subtracted from zero gives its negation.

$$580x-11x^{2}=-4800$$

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.

$$-11x^{2}+580x=-4800$$

Divide both sides by -11.

$$\frac{-11x^{2}+580x}{-11}=-\frac{4800}{-11}$$

Dividing by -11 undoes the multiplication by -11.

$$x^{2}+\frac{580}{-11}x=-\frac{4800}{-11}$$

Divide 580 by -11.

$$x^{2}-\frac{580}{11}x=-\frac{4800}{-11}$$

Divide -4800 by -11.

$$x^{2}-\frac{580}{11}x=\frac{4800}{11}$$

Divide -\frac{580}{11}, the coefficient of the x term, by 2 to get -\frac{290}{11}. Then add the square of -\frac{290}{11} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{580}{11}x+\left(-\frac{290}{11}\right)^{2}=\frac{4800}{11}+\left(-\frac{290}{11}\right)^{2}$$

Square -\frac{290}{11} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{580}{11}x+\frac{84100}{121}=\frac{4800}{11}+\frac{84100}{121}$$

Add \frac{4800}{11} to \frac{84100}{121} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{580}{11}x+\frac{84100}{121}=\frac{136900}{121}$$

Factor x^{2}-\frac{580}{11}x+\frac{84100}{121}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{290}{11}\right)^{2}=\frac{136900}{121}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{290}{11}\right)^{2}}=\sqrt{\frac{136900}{121}}$$

Simplify.

$$x-\frac{290}{11}=\frac{370}{11} x-\frac{290}{11}=-\frac{370}{11}$$

Add \frac{290}{11} to both sides of the equation.

$$x=60 x=-\frac{80}{11}$$