3-(4x-6)/(6-4x)=2x-3

Solution for 3-(4x-6)/(6-4x)=2x-3 equation:

3-(4x-6)/(6-4x)=2x-3

We move all terms to the left:

3-(4x-6)/(6-4x)-(2x-3)=0


Domain of the equation: (6-4x)!=0

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

-4x!=-6

x!=-6/-4

x!=1+1/2

x∈R


We add all the numbers together, and all the variables

-(4x-6)/(-4x+6)-(2x-3)+3=0

We get rid of parentheses

-(4x-6)/(-4x+6)-2x+3+3=0

We multiply all the terms by the denominator


-(4x-6)-2x*(-4x+6)+3*(-4x+6)+3*(-4x+6)=0

We multiply parentheses

8x^2-(4x-6)-12x-12x-12x+18+18=0

We get rid of parentheses

8x^2-4x-12x-12x-12x+6+18+18=0

We add all the numbers together, and all the variables

8x^2-40x+42=0

a = 8; b = -40; c = +42;
Δ = b2-4ac
Δ = -402-4·8·42
Δ = 256
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{256}=16$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-40)-16}{2*8}=\frac{24}{16}
=1+1/2
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-40)+16}{2*8}=\frac{56}{16}
=3+1/2
$