n(n+1)=5(n+1)+16

Solution for n(n+1)=5(n+1)+16 equation:

n(n+1)=5(n+1)+16

We move all terms to the left:

n(n+1)-(5(n+1)+16)=0

We multiply parentheses

n^2+n-(5(n+1)+16)=0


We calculate terms in parentheses: -(5(n+1)+16), so:

5(n+1)+16

We multiply parentheses

5n+5+16

We add all the numbers together, and all the variables

5n+21

Back to the equation:

-(5n+21)


We get rid of parentheses

n^2+n-5n-21=0

We add all the numbers together, and all the variables

n^2-4n-21=0

a = 1; b = -4; c = -21;
Δ = b2-4ac
Δ = -42-4·1·(-21)
Δ = 100
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{100}=10$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-10}{2*1}=\frac{-6}{2}
=-3
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+10}{2*1}=\frac{14}{2}
=7
$