(5x-7)-(2x+3)/(6x+11)=8/3

Solution for (5x-7)-(2x+3)/(6x+11)=8/3 equation:

(5x-7)-(2x+3)/(6x+11)=8/3

We move all terms to the left:

(5x-7)-(2x+3)/(6x+11)-(8/3)=0


Domain of the equation: (6x+11)!=0

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

6x!=-11

x!=-11/6

x!=-1+5/6

x∈R


We add all the numbers together, and all the variables

(5x-7)-(2x+3)/(6x+11)-(+8/3)=0

We get rid of parentheses

5x-(2x+3)/(6x+11)-7-8/3=0

We calculate fractions


5x+(-6x-9)/(18x+33)+(-48x-88)/(18x+33)-7=0

We multiply all the terms by the denominator


5x*(18x+33)+(-6x-9)+(-48x-88)-7*(18x+33)=0

We multiply parentheses

90x^2+165x+(-6x-9)+(-48x-88)-126x-231=0

We get rid of parentheses

90x^2+165x-6x-48x-126x-9-88-231=0

We add all the numbers together, and all the variables

90x^2-15x-328=0

a = 90; b = -15; c = -328;
Δ = b2-4ac
Δ = -152-4·90·(-328)
Δ = 118305
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:

$\sqrt{\Delta}=\sqrt{118305}=\sqrt{9*13145}=\sqrt{9}*\sqrt{13145}=3\sqrt{13145}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-15)-3\sqrt{13145}}{2*90}=\frac{15-3\sqrt{13145}}{180}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-15)+3\sqrt{13145}}{2*90}=\frac{15+3\sqrt{13145}}{180}
$