1/3x+9=3x+2

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

1/3x+9=3x+2

We move all terms to the left:

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


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

-9x^2-6x+27x+1=0

We add all the numbers together, and all the variables

-9x^2+21x+1=0

a = -9; b = 21; c = +1;
Δ = b2-4ac
Δ = 212-4·(-9)·1
Δ = 477
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:

$\sqrt{\Delta}=\sqrt{477}=\sqrt{9*53}=\sqrt{9}*\sqrt{53}=3\sqrt{53}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(21)-3\sqrt{53}}{2*-9}=\frac{-21-3\sqrt{53}}{-18}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(21)+3\sqrt{53}}{2*-9}=\frac{-21+3\sqrt{53}}{-18}
$