x(x+6)/(2x+6)=40/3

Solution for x(x+6)/(2x+6)=40/3 equation:

x(x+6)/(2x+6)=40/3

We move all terms to the left:

x(x+6)/(2x+6)-(40/3)=0


Domain of the equation: (2x+6)!=0

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

2x!=-6

x!=-6/2

x!=-3

x∈R


We add all the numbers together, and all the variables

x(x+6)/(2x+6)-(+40/3)=0

We get rid of parentheses

x(x+6)/(2x+6)-40/3=0

We calculate fractions


(3x^2+18x)/(6x+18)+(-80x-240)/(6x+18)=0

We multiply all the terms by the denominator


(3x^2+18x)+(-80x-240)=0

We get rid of parentheses

3x^2+18x-80x-240=0

We add all the numbers together, and all the variables

3x^2-62x-240=0

a = 3; b = -62; c = -240;
Δ = b2-4ac
Δ = -622-4·3·(-240)
Δ = 6724
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{6724}=82$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-62)-82}{2*3}=\frac{-20}{6}
=-3+1/3
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-62)+82}{2*3}=\frac{144}{6}
=24
$