P=(100x-0.5x^2)-(60x+300)

Solution for P=(100x-0.5x^2)-(60x+300) equation:

=(100P-0.5P^2)-(60P+300)

We move all terms to the left:

-((100P-0.5P^2)-(60P+300))=0


We calculate terms in parentheses: -((100P-0.5P^2)-(60P+300)), so:

(100P-0.5P^2)-(60P+300)

We get rid of parentheses

-0.5P^2+100P-60P-300

We add all the numbers together, and all the variables

-0.5P^2+40P-300

Back to the equation:

-(-0.5P^2+40P-300)


We get rid of parentheses

0.5P^2-40P+300=0

a = 0.5; b = -40; c = +300;
Δ = b2-4ac
Δ = -402-4·0.5·300
Δ = 1000
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:

$\sqrt{\Delta}=\sqrt{1000}=\sqrt{100*10}=\sqrt{100}*\sqrt{10}=10\sqrt{10}$
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-40)-10\sqrt{10}}{2*0.5}=\frac{40-10\sqrt{10}}{1}
$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-40)+10\sqrt{10}}{2*0.5}=\frac{40+10\sqrt{10}}{1}
$