The given equation is
P(q) = q(65 - 5q)
By opening the parentheses, we have
P(q) = 65q - 5q^2
If the average profit is $150 per quart, then P(q) = 150
The equation would be
150 = 65q - 5q^2
By rearranging it, we have
5q^2 - 65q + 150 = 0
Dividing through by 5, we have
q^2 - 13q + 30 = 0
To factorise, we would find two terms such that their sum or difference is - 13q and their product is 30q^2. The tare erms are - 10q and - 3q. Thus, we have
q^2 - 10q - 3q + 30 = 0
q(q - 10) - 3(q - 10) = 0
(q - 10)(q - 3) = 0
q - 10 = 0 or q - 3 = 0
q = 10 or q = 3
You must order 3 quarts or 10 quarts of the specialized metallic paints for the average profit to be $150 per quart