409=-0.5x^2+36x-185

Solution for 409=-0.5x^2+36x-185 equation:

409=-0.5x^2+36x-185

We move all terms to the left:

409-(-0.5x^2+36x-185)=0

We get rid of parentheses

0.5x^2-36x+185+409=0

We add all the numbers together, and all the variables

0.5x^2-36x+594=0

a = 0.5; b = -36; c = +594;
Δ = b2-4ac
Δ = -362-4·0.5·594
Δ = 108
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:

$\sqrt{\Delta}=\sqrt{108}=\sqrt{36*3}=\sqrt{36}*\sqrt{3}=6\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-36)-6\sqrt{3}}{2*0.5}=\frac{36-6\sqrt{3}}{1}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-36)+6\sqrt{3}}{2*0.5}=\frac{36+6\sqrt{3}}{1}
$