P(75)=-0.5x^2+400x-5000

Solution for P(75)=-0.5x^2+400x-5000 equation:

(75)=-0.5P^2+400P-5000

We move all terms to the left:

(75)-(-0.5P^2+400P-5000)=0

We get rid of parentheses

0.5P^2-400P+5000+75=0

We add all the numbers together, and all the variables

0.5P^2-400P+5075=0

a = 0.5; b = -400; c = +5075;
Δ = b2-4ac
Δ = -4002-4·0.5·5075
Δ = 149850
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:

$\sqrt{\Delta}=\sqrt{149850}=\sqrt{2025*74}=\sqrt{2025}*\sqrt{74}=45\sqrt{74}$
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-400)-45\sqrt{74}}{2*0.5}=\frac{400-45\sqrt{74}}{1}
$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-400)+45\sqrt{74}}{2*0.5}=\frac{400+45\sqrt{74}}{1}
$