-6n^2-6n+19=0

Solution for -6n^2-6n+19=0 equation:

-6n^2-6n+19=0

a = -6; b = -6; c = +19;
Δ = b2-4ac
Δ = -62-4·(-6)·19
Δ = 492
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:

$\sqrt{\Delta}=\sqrt{492}=\sqrt{4*123}=\sqrt{4}*\sqrt{123}=2\sqrt{123}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-2\sqrt{123}}{2*-6}=\frac{6-2\sqrt{123}}{-12}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+2\sqrt{123}}{2*-6}=\frac{6+2\sqrt{123}}{-12}
$