(240)/(x+20)+(240)/(x)=8

Solution for (240)/(x+20)+(240)/(x)=8 equation:

(240)/(x+20)+(240)/(x)=8

We move all terms to the left:

(240)/(x+20)+(240)/(x)-(8)=0


Domain of the equation: (x+20)!=0

We move all terms containing x to the left, all other terms to the right

x!=-20

x∈R



Domain of the equation: x!=0

x∈R


We calculate fractions


240x/(x^2+20x)+(240x+4800)/(x^2+20x)-8=0

We multiply all the terms by the denominator


240x+(240x+4800)-8*(x^2+20x)=0

We multiply parentheses

-8x^2+240x+(240x+4800)-160x=0

We get rid of parentheses

-8x^2+240x+240x-160x+4800=0

We add all the numbers together, and all the variables

-8x^2+320x+4800=0

a = -8; b = 320; c = +4800;
Δ = b2-4ac
Δ = 3202-4·(-8)·4800
Δ = 256000
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{256000}=\sqrt{25600*10}=\sqrt{25600}*\sqrt{10}=160\sqrt{10}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(320)-160\sqrt{10}}{2*-8}=\frac{-320-160\sqrt{10}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(320)+160\sqrt{10}}{2*-8}=\frac{-320+160\sqrt{10}}{-16}
$