(2x+5)+1/(3x-33)+1=1

Solution for (2x+5)+1/(3x-33)+1=1 equation:

(2x+5)+1/(3x-33)+1=1

We move all terms to the left:

(2x+5)+1/(3x-33)+1-(1)=0


Domain of the equation: (3x-33)!=0

We move all terms containing x to the left, all other terms to the right

3x!=33

x!=33/3

x!=11

x∈R


We add all the numbers together, and all the variables

(2x+5)+1/(3x-33)=0

We get rid of parentheses

2x+1/(3x-33)+5=0

We multiply all the terms by the denominator


2x*(3x-33)+5*(3x-33)+1=0

We multiply parentheses

6x^2-66x+15x-165+1=0

We add all the numbers together, and all the variables

6x^2-51x-164=0

a = 6; b = -51; c = -164;
Δ = b2-4ac
Δ = -512-4·6·(-164)
Δ = 6537
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-51)-\sqrt{6537}}{2*6}=\frac{51-\sqrt{6537}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-51)+\sqrt{6537}}{2*6}=\frac{51+\sqrt{6537}}{12}
$