1/2x+3=9-6x

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

1/2x+3=9-6x

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

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

We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

12x^2-18x+6x+1=0

We add all the numbers together, and all the variables

12x^2-12x+1=0

a = 12; b = -12; c = +1;
Δ = b2-4ac
Δ = -122-4·12·1
Δ = 96
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:

$\sqrt{\Delta}=\sqrt{96}=\sqrt{16*6}=\sqrt{16}*\sqrt{6}=4\sqrt{6}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-4\sqrt{6}}{2*12}=\frac{12-4\sqrt{6}}{24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+4\sqrt{6}}{2*12}=\frac{12+4\sqrt{6}}{24}
$