1/3x-3=2x+1

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

1/3x-3=2x+1

We move all terms to the left:

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


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

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

a = -6; b = -12; c = +1;
Δ = b2-4ac
Δ = -122-4·(-6)·1
Δ = 168
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:

$\sqrt{\Delta}=\sqrt{168}=\sqrt{4*42}=\sqrt{4}*\sqrt{42}=2\sqrt{42}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-2\sqrt{42}}{2*-6}=\frac{12-2\sqrt{42}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+2\sqrt{42}}{2*-6}=\frac{12+2\sqrt{42}}{-12}
$