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

Solution for 12(x-2)+3x=1/2x+6)+2 equation:

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

We move all terms to the left:

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


Domain of the equation: 2x+6)!=0

x∈R


We add all the numbers together, and all the variables

3x+12(x-2)-(1/2x+6)=0

We multiply parentheses

3x+12x-(1/2x+6)-24=0

We get rid of parentheses

3x+12x-1/2x-6-24=0

We multiply all the terms by the denominator


3x*2x+12x*2x-6*2x-24*2x-1=0

Wy multiply elements

6x^2+24x^2-12x-48x-1=0

We add all the numbers together, and all the variables

30x^2-60x-1=0

a = 30; b = -60; c = -1;
Δ = b2-4ac
Δ = -602-4·30·(-1)
Δ = 3720
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:

$\sqrt{\Delta}=\sqrt{3720}=\sqrt{4*930}=\sqrt{4}*\sqrt{930}=2\sqrt{930}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-60)-2\sqrt{930}}{2*30}=\frac{60-2\sqrt{930}}{60}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-60)+2\sqrt{930}}{2*30}=\frac{60+2\sqrt{930}}{60}
$