3(x+3)-1/2x=19

Solution for 3(x+3)-1/2x=19 equation:

3(x+3)-1/2x=19

We move all terms to the left:

3(x+3)-1/2x-(19)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We multiply parentheses

3x-1/2x+9-19=0

We multiply all the terms by the denominator


3x*2x+9*2x-19*2x-1=0

Wy multiply elements

6x^2+18x-38x-1=0

We add all the numbers together, and all the variables

6x^2-20x-1=0

a = 6; b = -20; c = -1;
Δ = b2-4ac
Δ = -202-4·6·(-1)
Δ = 424
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:

$\sqrt{\Delta}=\sqrt{424}=\sqrt{4*106}=\sqrt{4}*\sqrt{106}=2\sqrt{106}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-20)-2\sqrt{106}}{2*6}=\frac{20-2\sqrt{106}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-20)+2\sqrt{106}}{2*6}=\frac{20+2\sqrt{106}}{12}
$