1/2x-6+3/2x=3x-2

Solution for 1/2x-6+3/2x=3x-2 equation:

1/2x-6+3/2x=3x-2

We move all terms to the left:

1/2x-6+3/2x-(3x-2)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x+3/2x-3x+2-6=0

We multiply all the terms by the denominator


-3x*2x+2*2x-6*2x+1+3=0

We add all the numbers together, and all the variables

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

Wy multiply elements

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

We add all the numbers together, and all the variables

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

a = -6; b = -8; c = +4;
Δ = b2-4ac
Δ = -82-4·(-6)·4
Δ = 160
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:

$\sqrt{\Delta}=\sqrt{160}=\sqrt{16*10}=\sqrt{16}*\sqrt{10}=4\sqrt{10}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-4\sqrt{10}}{2*-6}=\frac{8-4\sqrt{10}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+4\sqrt{10}}{2*-6}=\frac{8+4\sqrt{10}}{-12}
$