2/3x-5=3x+9

Solution for 2/3x-5=3x+9 equation:

2/3x-5=3x+9

We move all terms to the left:

2/3x-5-(3x+9)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

2/3x-3x-9-5=0

We multiply all the terms by the denominator


-3x*3x-9*3x-5*3x+2=0

Wy multiply elements

-9x^2-27x-15x+2=0

We add all the numbers together, and all the variables

-9x^2-42x+2=0

a = -9; b = -42; c = +2;
Δ = b2-4ac
Δ = -422-4·(-9)·2
Δ = 1836
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:

$\sqrt{\Delta}=\sqrt{1836}=\sqrt{36*51}=\sqrt{36}*\sqrt{51}=6\sqrt{51}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-42)-6\sqrt{51}}{2*-9}=\frac{42-6\sqrt{51}}{-18}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-42)+6\sqrt{51}}{2*-9}=\frac{42+6\sqrt{51}}{-18}
$