Question

A bike manufacturer can predict profits, P, from a new sports bike using the quadratic function P(x) = −100x2 + 46,000x − 2,100,000, where x is the price of the bike. At what prices will the company make $0 in profit?

Answer 1

At $51.39 and $408.61, the company will make $0 in profit.

Explanation:

Hi there!

Let´s write the function:

P(x) = -100 x² + 46,000 x - 2,100,000

We have to find the value of x at which P(x) = 0:

0 = -100 x² + 46,000 x - 2,100,000

Using the quadratic formula, we can solve this quadratic equation:

a = -100

b = 46,000

c = -2,100,000

x = [-b ±√(b² - 4ac)]/2a

x = [-46,000 ± √(46,000² - 4(-100)(-2,100,000)] / 2(-100)

x = (-46,000 ± 35,721.14)/-200

x₁ = (-46,000 + 35,721.14)/-200

x₁ = 51.39

x₂ = (-46,000 - 35,721.14)/-200

x₂ = 408.61

Then at $51.39 and $408.61 the company will make $0 in profit.