2=3x-9/4x-6

Solution for 2=3x-9/4x-6 equation:

2=3x-9/4x-6

We move all terms to the left:

2-(3x-9/4x-6)=0


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

x∈R


We get rid of parentheses

-3x+9/4x+6+2=0

We multiply all the terms by the denominator


-3x*4x+6*4x+2*4x+9=0

Wy multiply elements

-12x^2+24x+8x+9=0

We add all the numbers together, and all the variables

-12x^2+32x+9=0

a = -12; b = 32; c = +9;
Δ = b2-4ac
Δ = 322-4·(-12)·9
Δ = 1456
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:

$\sqrt{\Delta}=\sqrt{1456}=\sqrt{16*91}=\sqrt{16}*\sqrt{91}=4\sqrt{91}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(32)-4\sqrt{91}}{2*-12}=\frac{-32-4\sqrt{91}}{-24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(32)+4\sqrt{91}}{2*-12}=\frac{-32+4\sqrt{91}}{-24}
$