P(x)=x(600-3x)-2(600-3x)-180

Solution for P(x)=x(600-3x)-2(600-3x)-180 equation:

(P)=P(600-3P)-2(600-3P)-180

We move all terms to the left:

(P)-(P(600-3P)-2(600-3P)-180)=0

We add all the numbers together, and all the variables

P-(P(-3P+600)-2(-3P+600)-180)=0


We calculate terms in parentheses: -(P(-3P+600)-2(-3P+600)-180), so:

P(-3P+600)-2(-3P+600)-180

We multiply parentheses

-3P^2+600P+6P-1200-180

We add all the numbers together, and all the variables

-3P^2+606P-1380

Back to the equation:

-(-3P^2+606P-1380)


We get rid of parentheses

3P^2-606P+P+1380=0

We add all the numbers together, and all the variables

3P^2-605P+1380=0

a = 3; b = -605; c = +1380;
Δ = b2-4ac
Δ = -6052-4·3·1380
Δ = 349465
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{-(-605)-\sqrt{349465}}{2*3}=\frac{605-\sqrt{349465}}{6}
$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-605)+\sqrt{349465}}{2*3}=\frac{605+\sqrt{349465}}{6}
$