6+1/2x=3x-9

Solution for 6+1/2x=3x-9 equation:

6+1/2x=3x-9

We move all terms to the left:

6+1/2x-(3x-9)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-3x+9+6=0

We multiply all the terms by the denominator


-3x*2x+9*2x+6*2x+1=0

Wy multiply elements

-6x^2+18x+12x+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}
$