38x-x^2-180.5=0

Solution for 38x-x^2-180.5=0 equation:

38x-x^2-180.5=0

We add all the numbers together, and all the variables

-1x^2+38x-180.5=0

a = -1; b = 38; c = -180.5;
Δ = b2-4ac
Δ = 382-4·(-1)·(-180.5)
Δ = 722
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:

$\sqrt{\Delta}=\sqrt{722}=\sqrt{361*2}=\sqrt{361}*\sqrt{2}=19\sqrt{2}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(38)-19\sqrt{2}}{2*-1}=\frac{-38-19\sqrt{2}}{-2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(38)+19\sqrt{2}}{2*-1}=\frac{-38+19\sqrt{2}}{-2}
$