P(x)=-x^2+48*x-512

Solution for P(x)=-x^2+48*x-512 equation:

(P)=-P^2+48P-512

We move all terms to the left:

(P)-(-P^2+48P-512)=0

We get rid of parentheses

P^2-48P+P+512=0

We add all the numbers together, and all the variables

P^2-47P+512=0

a = 1; b = -47; c = +512;
Δ = b2-4ac
Δ = -472-4·1·512
Δ = 161
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-47)-\sqrt{161}}{2*1}=\frac{47-\sqrt{161}}{2}
$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-47)+\sqrt{161}}{2*1}=\frac{47+\sqrt{161}}{2}
$