P(n)=n(120-n)-50n

Solution for P(n)=n(120-n)-50n equation:

(P)=P(120-P)-50P

We move all terms to the left:

(P)-(P(120-P)-50P)=0

We add all the numbers together, and all the variables

P-(P(-1P+120)-50P)=0


We calculate terms in parentheses: -(P(-1P+120)-50P), so:

P(-1P+120)-50P

We add all the numbers together, and all the variables

-50P+P(-1P+120)

We multiply parentheses

-1P^2-50P+120P

We add all the numbers together, and all the variables

-1P^2+70P

Back to the equation:

-(-1P^2+70P)


We get rid of parentheses

1P^2-70P+P=0

We add all the numbers together, and all the variables

P^2-69P=0

a = 1; b = -69; c = 0;
Δ = b2-4ac
Δ = -692-4·1·0
Δ = 4761
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{4761}=69$
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-69)-69}{2*1}=\frac{0}{2}
=0
$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-69)+69}{2*1}=\frac{138}{2}
=69
$