(2x+3)-(5x-7)/6x+11=8/15

Solution for (2x+3)-(5x-7)/6x+11=8/15 equation:

(2x+3)-(5x-7)/6x+11=8/15

We move all terms to the left:

(2x+3)-(5x-7)/6x+11-(8/15)=0


Domain of the equation: 6x!=0

x!=0/6

x!=0

x∈R


We add all the numbers together, and all the variables

(2x+3)-(5x-7)/6x+11-(+8/15)=0

We get rid of parentheses

2x-(5x-7)/6x+3+11-8/15=0

We calculate fractions


2x+(-75x+105)/90x+(-48x)/90x+3+11=0

We add all the numbers together, and all the variables

2x+(-75x+105)/90x+(-48x)/90x+14=0

We multiply all the terms by the denominator


2x*90x+(-75x+105)+(-48x)+14*90x=0

Wy multiply elements

180x^2+(-75x+105)+(-48x)+1260x=0

We get rid of parentheses

180x^2-75x-48x+1260x+105=0

We add all the numbers together, and all the variables

180x^2+1137x+105=0

a = 180; b = 1137; c = +105;
Δ = b2-4ac
Δ = 11372-4·180·105
Δ = 1217169
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:

$\sqrt{\Delta}=\sqrt{1217169}=\sqrt{9*135241}=\sqrt{9}*\sqrt{135241}=3\sqrt{135241}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1137)-3\sqrt{135241}}{2*180}=\frac{-1137-3\sqrt{135241}}{360}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1137)+3\sqrt{135241}}{2*180}=\frac{-1137+3\sqrt{135241}}{360}
$