A Quadratic Word Problem

Date: 07/27/99 at 01:00:17
From: Robbie Coleman
Subject: A Quadratic Word Problem

If the robbery victimization rate per 1000 persons for African-
Americans is increased by two and then decreased by 6, the product of 
these two numbers is 180. What is the rate?

My attempt:

Let r = rate.
Now,
   (r + 2 - 6)r = 180
       r(r - 4) = 180
       r^2 - 4r = 180
 r^2 - 4r - 180 = 0

That is as far as I have gotten.

Date: 07/27/99 at 12:44:53
From: Doctor Rick
Subject: Re: A Quadratic Word Problem

The problem is not very clearly stated; I do not think it means what 
you took it to mean. I think "the product of these two numbers" means 
the product of the rate increased by 2 and the rate decreased by 6. 
See what quadratic equation you get if you interpret the question that 
way.

The new quadratic equation can be solved by factorization, but the 
quadratic equation you got is best solved using the quadratic formula. 
This is how I would solve either quadratic equation, so let us go over 
this method.

If you have a quadratic equation of the form

  ax^2 + bx + c = 0

where a, b, and c are three constant numbers, then the two solutions 
for x are

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

In your quadratic equation,

  r^2 - 4r - 180 = 0

the constant coefficients are

  a = 1
  b = -4
  c = -180

Plugging these values for a, b, and c into the quadratic formula, we 
get

  r = (4 + or - sqrt((-4)^2 + 4*180))/2
    = (4 + or - sqrt(736))/2
    = (4 + or - 27.12932)/2
    = 31.12932/2 or -23.12932/2
    = 15.56466 or -11.56466

Since a robbery rate must be non-negative, we can eliminate the second 
(negative) solution, and the answer is 15.56466.

Now you can solve the problem as I interpret it, which is easier (it 
has integer solutions).

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/