6x-9=1/3x+6

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

6x-9=1/3x+6

We move all terms to the left:

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


Domain of the equation: 3x+6)!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

18x^2-18x-27x-1=0

We add all the numbers together, and all the variables

18x^2-45x-1=0

a = 18; b = -45; c = -1;
Δ = b2-4ac
Δ = -452-4·18·(-1)
Δ = 2097
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:

$\sqrt{\Delta}=\sqrt{2097}=\sqrt{9*233}=\sqrt{9}*\sqrt{233}=3\sqrt{233}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-45)-3\sqrt{233}}{2*18}=\frac{45-3\sqrt{233}}{36}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-45)+3\sqrt{233}}{2*18}=\frac{45+3\sqrt{233}}{36}
$