2n+14=2n(n+1)

Solution for 2n+14=2n(n+1) equation:

2n+14=2n(n+1)

We move all terms to the left:

2n+14-(2n(n+1))=0


We calculate terms in parentheses: -(2n(n+1)), so:

2n(n+1)

We multiply parentheses

2n^2+2n

Back to the equation:

-(2n^2+2n)


We get rid of parentheses

-2n^2+2n-2n+14=0

We add all the numbers together, and all the variables

-2n^2+14=0

a = -2; b = 0; c = +14;
Δ = b2-4ac
Δ = 02-4·(-2)·14
Δ = 112
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{112}=\sqrt{16*7}=\sqrt{16}*\sqrt{7}=4\sqrt{7}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-4\sqrt{7}}{2*-2}=\frac{0-4\sqrt{7}}{-4}
=-\frac{4\sqrt{7}}{-4}
=-\frac{\sqrt{7}}{-1}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+4\sqrt{7}}{2*-2}=\frac{0+4\sqrt{7}}{-4}
=\frac{4\sqrt{7}}{-4}
=\frac{\sqrt{7}}{-1}
$