Quadratic Equation - Small Business and Profit

Question

Perry owns a small business selling bags of coffee beans online. If he sells coffee bean bags for p dollars per bag, then he expects to earn a profit modeled by the function

P(p) = 4p^2 + 120p - 500

What are the two prices that would lead to a profit of zero dollars?

Answer 1

To figure out the price that would lead to a profit of zero dollars, we set our equation:

4p^2 + 120p - 500 = 0.

We notice that all terms are divisible by 4, so we can divide both sides of the equation by 4 to get:

p^2 + 30p - 125 = 0.

We cannot factor this equation regularly as there are no rational numbers whose sum is 30 and product is -125. So, we must use the quadratic formula:

p = (-b +/- sqrt(b^2 - 4ac)) / 2a

In our case a = 1, b = 30, and c = -125. So:

p = (-30 +/- sqrt(900 - 4(1)(-125)) / 2(1). Simplifying this, we get that:

p = (-30 +/- sqrt(900 + 500)) / 2 , so:

p = (-30 +/- sqrt(1400)) / 2. The square root of 1400 can be rewritten as 10* sqrt(14). So:

p = (-30 +/- (10sqrt(14)) / 2. Dividing by 2, we can simplify to:

p = -15 +/- 5sqrt(14). Taking 5 as the common factor, we can write:

p = 5(-3 +/- sqrt(14)), so:

p = 5(-3 + sqrt(14)), and p = 5(-3 - sqrt(14)). We can rewrite these as:

p = 5(sqrt(14) - 3), and p = -5(3 + sqrt(14)). These are the two prices that would lead to a $0 dollar profit.