78=19n-n^2

Solution for 78=19n-n^2 equation:

78=19n-n^2

We move all terms to the left:

78-(19n-n^2)=0

We get rid of parentheses

n^2-19n+78=0

a = 1; b = -19; c = +78;
Δ = b2-4ac
Δ = -192-4·1·78
Δ = 49
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{49}=7$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-19)-7}{2*1}=\frac{12}{2}
=6
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-19)+7}{2*1}=\frac{26}{2}
=13
$