3*x-2=-1/3x+3

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

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

We move all terms to the left:

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


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

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

9x^2-15x+1=0

a = 9; b = -15; c = +1;
Δ = b2-4ac
Δ = -152-4·9·1
Δ = 189
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:

$\sqrt{\Delta}=\sqrt{189}=\sqrt{9*21}=\sqrt{9}*\sqrt{21}=3\sqrt{21}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-15)-3\sqrt{21}}{2*9}=\frac{15-3\sqrt{21}}{18}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-15)+3\sqrt{21}}{2*9}=\frac{15+3\sqrt{21}}{18}
$