P=r(150-1.5r)-20(150-1.5r)-1,500

Solution for P=r(150-1.5r)-20(150-1.5r)-1,500 equation:

=P(150-1.5P)-20(150-1.5P)-1.500

We move all terms to the left:

-(P(150-1.5P)-20(150-1.5P)-1.500)=0

We add all the numbers together, and all the variables

-(P(-1.5P+150)-20(-1.5P+150)-1.500)=0


We calculate terms in parentheses: -(P(-1.5P+150)-20(-1.5P+150)-1.500), so:

P(-1.5P+150)-20(-1.5P+150)-1.500

We add all the numbers together, and all the variables

P(-1.5P+150)-20(-1.5P+150)-1.5

We multiply parentheses

-1P^2+150P+20P-3000-1.5

We add all the numbers together, and all the variables

-1P^2+170P-3001.5

Back to the equation:

-(-1P^2+170P-3001.5)


We get rid of parentheses

1P^2-170P+3001.5=0

We add all the numbers together, and all the variables

P^2-170P+3001.5=0

a = 1; b = -170; c = +3001.5;
Δ = b2-4ac
Δ = -1702-4·1·3001.5
Δ = 16894
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{-(-170)-\sqrt{16894}}{2*1}=\frac{170-\sqrt{16894}}{2}
$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-170)+\sqrt{16894}}{2*1}=\frac{170+\sqrt{16894}}{2}
$