x/(5x+12)=2x-3

Solution for x/(5x+12)=2x-3 equation:

x/(5x+12)=2x-3

We move all terms to the left:

x/(5x+12)-(2x-3)=0


Domain of the equation: (5x+12)!=0

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

5x!=-12

x!=-12/5

x!=-2+2/5

x∈R


We get rid of parentheses

x/(5x+12)-2x+3=0

We multiply all the terms by the denominator


x-2x*(5x+12)+3*(5x+12)=0

We multiply parentheses

-10x^2+x-24x+15x+36=0

We add all the numbers together, and all the variables

-10x^2-8x+36=0

a = -10; b = -8; c = +36;
Δ = b2-4ac
Δ = -82-4·(-10)·36
Δ = 1504
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:

$\sqrt{\Delta}=\sqrt{1504}=\sqrt{16*94}=\sqrt{16}*\sqrt{94}=4\sqrt{94}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-4\sqrt{94}}{2*-10}=\frac{8-4\sqrt{94}}{-20}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+4\sqrt{94}}{2*-10}=\frac{8+4\sqrt{94}}{-20}
$