Quadratic Model

Date: 12/03/2001 at 22:35:31
From: Erin Miller
Subject: Quadratic model

The quadratic model S = -17t^2 + 45t + 2570 approximates the number of 
suicides by firearms in the U.S. for women, where t represents the 
number of years after 1985. In what year were there 2370 female 
suicides by firearms?

I subtracted 2370 from 2570 to set the equation to 0. I then tried 
to solve for t by plugging the numbers into the quadratic equation, 
but the numbers are too large to work with. I wasn't able to simplify 
the problem before plugging it into the equation, so I'm stuck. Please 
help.

Thanks,
Erin

Date: 12/04/2001 at 02:29:59
From: Doctor Jeremiah
Subject: Re: Quadratic model

Hi Erin,

To answer this you need to do what is called "completing the square."

        2370 = -17t^2 + 45t + 2570
 2370 - 2570 = -17t^2 + 45t + 2570 - 2570
        -200 = -17t^2 + 45t
    -200/-17 = (-17t^2 + 45t)/-17
    -200/-17 = -17t^2/-17 + 45t/-17
      200/17 = t^2 + (-45/17)t

Now we need to know what to add to the right side to make a perfect 
square (something where the entire right side is squared). In a 
perfect square:

  (t+a)^2 = t^2 + 2at + a^2

So that means that 2at is the equivalent of (45/-17)t, which would 
mean that 2a = -45/17. This means that:

     2a = -45/17
   2a/2 = (-45/17)/2
      a = -45/34

And a^2 = (-45/34)^2

To make a perfect square we need to add a^2 to the right side of the 
equation. But since we must do all operations to both sides, we will 
add (-45/34)^2 to both sides:

              200/17 = t^2 + (-45/17)t
 200/17 + (-45/34)^2 = t^2 + (-45/17)t + (-45/34)^2

Now the right side is a perfect square of the form (t+a)^2, and since 
a = -45/34 we have:

 200/17 + (-45/34)^2 = t^2 + (-45/17)t + (-45/34)^2
 200/17 + (-45/34)^2 = (t + -45/34)^2
 200/17 + (-45/34)^2 = (t - 45/34)^2

Now we can take the square root of both sides to get:

 sqrt( 200/17 + (-45/34)^2 ) = sqrt( (t - 45/34)^2 )
 sqrt( 200/17 + (-45/34)^2 ) = t - 45/34

And then we can add 45/34 to both sides to get:

 +- sqrt( 200/17 + (-45/34)^2 ) = t - 45/34

Remember that there are two answers for the square root (the + answer 
and the - answer):

 45/34 +- sqrt( 200/17 + (-45/34)^2 ) = t - 45/34 + 45/34
 45/34 +- sqrt( 200/17 + (-45/34)^2 ) = t

And now we have something equal to t, so if we simplify it we will 
have the answer we want. You should be able to do the simplification
just with algebra. In the end there are two answers. The positive 
answer is the one they are asking for.

Since t is the number of years after 1985, be sure to add t to 1985 to 
get the year.

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