2/3x+2=3x-5

Solution for 2/3x+2=3x-5 equation:

2/3x+2=3x-5

We move all terms to the left:

2/3x+2-(3x-5)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

2/3x-3x+5+2=0

We multiply all the terms by the denominator


-3x*3x+5*3x+2*3x+2=0

Wy multiply elements

-9x^2+15x+6x+2=0

We add all the numbers together, and all the variables

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

a = -9; b = 21; c = +2;
Δ = b2-4ac
Δ = 212-4·(-9)·2
Δ = 513
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:

$\sqrt{\Delta}=\sqrt{513}=\sqrt{9*57}=\sqrt{9}*\sqrt{57}=3\sqrt{57}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(21)-3\sqrt{57}}{2*-9}=\frac{-21-3\sqrt{57}}{-18}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(21)+3\sqrt{57}}{2*-9}=\frac{-21+3\sqrt{57}}{-18}
$