SOLUTION: The number of coffee shops in a certain country can be modeled by the quadratic function f(x)=111x^2+680x+4486 where f(x) is the number of the coffee shops and x is the number of y

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Question 633133: The number of coffee shops in a certain country can be modeled by the quadratic function f(x)=111x^2+680x+4486 where f(x) is the number of the coffee shops and x is the number of years after 2000. Two part answer - part a) the number of coffee shops in 2004 was? and b) the year in which the number of coffee shops will be 20,000 is?
I've tried trying applying the quadratic formula for the first part, but I don't understand the two parts they want answered. Thank you for your help!
Answer by graphmatics(170) About Me  (Show Source):
You can put this solution on YOUR website!
To solve part a) compute f(4)=111*4^2+680*4+4486=1776+2720+4486=8982. To solve part b) find x from f(x)=20000. 111*x^2+680*x+4486=20000 111*x^2+680*x-15514=0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 111x%5E2%2B680x%2B-15514+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28680%29%5E2-4%2A111%2A-15514=7350616.

Discriminant d=7350616 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-680%2B-sqrt%28+7350616+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28680%29%2Bsqrt%28+7350616+%29%29%2F2%5C111+=+9.14955832196591
x%5B2%5D+=+%28-%28680%29-sqrt%28+7350616+%29%29%2F2%5C111+=+-15.275684448092

Quadratic expression 111x%5E2%2B680x%2B-15514 can be factored:
111x%5E2%2B680x%2B-15514+=+111%28x-9.14955832196591%29%2A%28x--15.275684448092%29
Again, the answer is: 9.14955832196591, -15.275684448092. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+111%2Ax%5E2%2B680%2Ax%2B-15514+%29