3x-9=1/2x+12

Solution for 3x-9=1/2x+12 equation:

3x-9=1/2x+12

We move all terms to the left:

3x-9-(1/2x+12)=0


Domain of the equation: 2x+12)!=0

x∈R


We get rid of parentheses

3x-1/2x-12-9=0

We multiply all the terms by the denominator


3x*2x-12*2x-9*2x-1=0

Wy multiply elements

6x^2-24x-18x-1=0

We add all the numbers together, and all the variables

6x^2-42x-1=0

a = 6; b = -42; c = -1;
Δ = b2-4ac
Δ = -422-4·6·(-1)
Δ = 1788
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:

$\sqrt{\Delta}=\sqrt{1788}=\sqrt{4*447}=\sqrt{4}*\sqrt{447}=2\sqrt{447}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-42)-2\sqrt{447}}{2*6}=\frac{42-2\sqrt{447}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-42)+2\sqrt{447}}{2*6}=\frac{42+2\sqrt{447}}{12}
$