P(x)=-0.5x^2+8x-24

Solution for P(x)=-0.5x^2+8x-24 equation:

(P)=-0.5P^2+8P-24

We move all terms to the left:

(P)-(-0.5P^2+8P-24)=0

We get rid of parentheses

0.5P^2-8P+P+24=0

We add all the numbers together, and all the variables

0.5P^2-7P+24=0

a = 0.5; b = -7; c = +24;
Δ = b2-4ac
Δ = -72-4·0.5·24
Δ = 1
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{1}=1$
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-7)-1}{2*0.5}=\frac{6}{1}
=6
$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-7)+1}{2*0.5}=\frac{8}{1}
=8
$