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

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

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

We move all terms to the left:

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


Domain of the equation: 2x+2)!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

6x^2+4x-12x+1=0

We add all the numbers together, and all the variables

6x^2-8x+1=0

a = 6; b = -8; c = +1;
Δ = b2-4ac
Δ = -82-4·6·1
Δ = 40
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:

$\sqrt{\Delta}=\sqrt{40}=\sqrt{4*10}=\sqrt{4}*\sqrt{10}=2\sqrt{10}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-2\sqrt{10}}{2*6}=\frac{8-2\sqrt{10}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+2\sqrt{10}}{2*6}=\frac{8+2\sqrt{10}}{12}
$