(5x+10)/(8x-20)=x

Solution for (5x+10)/(8x-20)=x equation:

(5x+10)/(8x-20)=x

We move all terms to the left:

(5x+10)/(8x-20)-(x)=0


Domain of the equation: (8x-20)!=0

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

8x!=20

x!=20/8

x!=2+1/2

x∈R


We add all the numbers together, and all the variables

-1x+(5x+10)/(8x-20)=0

We multiply all the terms by the denominator


-1x*(8x-20)+(5x+10)=0

We multiply parentheses

-8x^2+20x+(5x+10)=0

We get rid of parentheses

-8x^2+20x+5x+10=0

We add all the numbers together, and all the variables

-8x^2+25x+10=0

a = -8; b = 25; c = +10;
Δ = b2-4ac
Δ = 252-4·(-8)·10
Δ = 945
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:

$\sqrt{\Delta}=\sqrt{945}=\sqrt{9*105}=\sqrt{9}*\sqrt{105}=3\sqrt{105}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(25)-3\sqrt{105}}{2*-8}=\frac{-25-3\sqrt{105}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(25)+3\sqrt{105}}{2*-8}=\frac{-25+3\sqrt{105}}{-16}
$