x^2=6x+19=0

Solution for x^2=6x+19=0 equation:

x^2=6x+19=0

We move all terms to the left:

x^2-(6x+19)=0

We get rid of parentheses

x^2-6x-19=0

a = 1; b = -6; c = -19;
Δ = b2-4ac
Δ = -62-4·1·(-19)
Δ = 112
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{112}=\sqrt{16*7}=\sqrt{16}*\sqrt{7}=4\sqrt{7}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-4\sqrt{7}}{2*1}=\frac{6-4\sqrt{7}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+4\sqrt{7}}{2*1}=\frac{6+4\sqrt{7}}{2}
$