P(x)=-x^2+10x-9

Solution for P(x)=-x^2+10x-9 equation:

(P)=-P^2+10P-9

We move all terms to the left:

(P)-(-P^2+10P-9)=0

We get rid of parentheses

P^2-10P+P+9=0

We add all the numbers together, and all the variables

P^2-9P+9=0

a = 1; b = -9; c = +9;
Δ = b2-4ac
Δ = -92-4·1·9
Δ = 45
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:

$\sqrt{\Delta}=\sqrt{45}=\sqrt{9*5}=\sqrt{9}*\sqrt{5}=3\sqrt{5}$
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-9)-3\sqrt{5}}{2*1}=\frac{9-3\sqrt{5}}{2}
$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-9)+3\sqrt{5}}{2*1}=\frac{9+3\sqrt{5}}{2}
$