P(x)=-x^2+10x+31

Solution for P(x)=-x^2+10x+31 equation:

(P)=-P^2+10P+31

We move all terms to the left:

(P)-(-P^2+10P+31)=0

We get rid of parentheses

P^2-10P+P-31=0

We add all the numbers together, and all the variables

P^2-9P-31=0

a = 1; b = -9; c = -31;
Δ = b2-4ac
Δ = -92-4·1·(-31)
Δ = 205
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{-(-9)-\sqrt{205}}{2*1}=\frac{9-\sqrt{205}}{2}
$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-9)+\sqrt{205}}{2*1}=\frac{9+\sqrt{205}}{2}
$