P=-x^2+66x+83

Solution for P=-x^2+66x+83 equation:

=-P^2+66P+83

We move all terms to the left:

-(-P^2+66P+83)=0

We get rid of parentheses

P^2-66P-83=0

a = 1; b = -66; c = -83;
Δ = b2-4ac
Δ = -662-4·1·(-83)
Δ = 4688
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{4688}=\sqrt{16*293}=\sqrt{16}*\sqrt{293}=4\sqrt{293}$
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-66)-4\sqrt{293}}{2*1}=\frac{66-4\sqrt{293}}{2}
$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-66)+4\sqrt{293}}{2*1}=\frac{66+4\sqrt{293}}{2}
$