3x+12=1/2x-3

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

3x+12=1/2x-3

We move all terms to the left:

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


Domain of the equation: 2x-3)!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

6x^2+30x-1=0

a = 6; b = 30; c = -1;
Δ = b2-4ac
Δ = 302-4·6·(-1)
Δ = 924
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:

$\sqrt{\Delta}=\sqrt{924}=\sqrt{4*231}=\sqrt{4}*\sqrt{231}=2\sqrt{231}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(30)-2\sqrt{231}}{2*6}=\frac{-30-2\sqrt{231}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(30)+2\sqrt{231}}{2*6}=\frac{-30+2\sqrt{231}}{12}
$