66=0.5n^2+0.5n

Solution for 66=0.5n^2+0.5n equation:

66=0.5n^2+0.5n

We move all terms to the left:

66-(0.5n^2+0.5n)=0

We get rid of parentheses

-0.5n^2-0.5n+66=0

a = -0.5; b = -0.5; c = +66;
Δ = b2-4ac
Δ = -0.52-4·(-0.5)·66
Δ = 132.25
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-0.5)-\sqrt{132.25}}{2*-0.5}=\frac{0.5-\sqrt{132.25}}{-1}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-0.5)+\sqrt{132.25}}{2*-0.5}=\frac{0.5+\sqrt{132.25}}{-1}
$