5(2+n)=3(n+6)n=

Solution for 5(2+n)=3(n+6)n= equation:

5(2+n)=3(n+6)n=

We move all terms to the left:

5(2+n)-(3(n+6)n)=0

We add all the numbers together, and all the variables

5(n+2)-(3(n+6)n)=0

We multiply parentheses

5n-(3(n+6)n)+10=0


We calculate terms in parentheses: -(3(n+6)n), so:

3(n+6)n

We multiply parentheses

3n^2+18n

Back to the equation:

-(3n^2+18n)


We get rid of parentheses

-3n^2+5n-18n+10=0

We add all the numbers together, and all the variables

-3n^2-13n+10=0

a = -3; b = -13; c = +10;
Δ = b2-4ac
Δ = -132-4·(-3)·10
Δ = 289
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{289}=17$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-13)-17}{2*-3}=\frac{-4}{-6}
=2/3
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-13)+17}{2*-3}=\frac{30}{-6}
=-5
$