22.5/2x-5=20/x+8

Solution for 22.5/2x-5=20/x+8 equation:

22.5/2x-5=20/x+8

We move all terms to the left:

22.5/2x-5-(20/x+8)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R



Domain of the equation: x+8)!=0

x∈R


We get rid of parentheses

22.5/2x-20/x-8-5=0

We calculate fractions


(22.5x)/2x^2+(-40x)/2x^2-8-5=0

We add all the numbers together, and all the variables

(+22.5x)/2x^2+(-40x)/2x^2-8-5=0

We add all the numbers together, and all the variables

(+22.5x)/2x^2+(-40x)/2x^2-13=0

We multiply all the terms by the denominator


(+22.5x)+(-40x)-13*2x^2=0

Wy multiply elements

-26x^2+(+22.5x)+(-40x)=0

We get rid of parentheses

-26x^2+22.5x-40x=0

We add all the numbers together, and all the variables

-26x^2-17.5x=0

a = -26; b = -17.5; c = 0;
Δ = b2-4ac
Δ = -17.52-4·(-26)·0
Δ = 306.25
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-17.5)-\sqrt{306.25}}{2*-26}=\frac{17.5-\sqrt{306.25}}{-52}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-17.5)+\sqrt{306.25}}{2*-26}=\frac{17.5+\sqrt{306.25}}{-52}
$