2x-(2x-3)/3x-(4x+5)=-1

Solution for 2x-(2x-3)/3x-(4x+5)=-1 equation:

2x-(2x-3)/3x-(4x+5)=-1

We move all terms to the left:

2x-(2x-3)/3x-(4x+5)-(-1)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We add all the numbers together, and all the variables

2x-(2x-3)/3x-(4x+5)+1=0

We get rid of parentheses

2x-(2x-3)/3x-4x-5+1=0

We multiply all the terms by the denominator


2x*3x-(2x-3)-4x*3x-5*3x+1*3x=0

Wy multiply elements

6x^2-12x^2-(2x-3)-15x+3x=0

We get rid of parentheses

6x^2-12x^2-2x-15x+3x+3=0

We add all the numbers together, and all the variables

-6x^2-14x+3=0

a = -6; b = -14; c = +3;
Δ = b2-4ac
Δ = -142-4·(-6)·3
Δ = 268
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:

$\sqrt{\Delta}=\sqrt{268}=\sqrt{4*67}=\sqrt{4}*\sqrt{67}=2\sqrt{67}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-14)-2\sqrt{67}}{2*-6}=\frac{14-2\sqrt{67}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-14)+2\sqrt{67}}{2*-6}=\frac{14+2\sqrt{67}}{-12}
$