2/3x+3=6x-3

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

2/3x+3=6x-3

We move all terms to the left:

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


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

-18x^2+9x+9x+2=0

We add all the numbers together, and all the variables

-18x^2+18x+2=0

a = -18; b = 18; c = +2;
Δ = b2-4ac
Δ = 182-4·(-18)·2
Δ = 468
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:

$\sqrt{\Delta}=\sqrt{468}=\sqrt{36*13}=\sqrt{36}*\sqrt{13}=6\sqrt{13}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(18)-6\sqrt{13}}{2*-18}=\frac{-18-6\sqrt{13}}{-36}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(18)+6\sqrt{13}}{2*-18}=\frac{-18+6\sqrt{13}}{-36}
$