Two Values of X

Date: 03/20/2002 at 00:13:28
From: Krystal
Subject: Word problem

Two numbers differ by 7 and the sum of their squares is 389. Find the 
two numbers. 

I set up the equation as 389 = (x) squared + (x-7) squared, but I 
can't get two true answers. Can you help me?

Date: 03/20/2002 at 01:45:57
From: Doctor Jeremiah
Subject: Re: Word problem

Hi Krystal,

I think the problem is that you are misinterpreting the answer that 
you are getting.

Your equation  "389 = (x) squared + (x-7) squared"  is correct.

        389 = (x) squared + (x-7) squared
        389 = (x) squared + (x) squared - 14x + 49
        389 = 2 (x) squared - 14x + 49
          0 = 2 (x) squared - 14x + 49 - 389
          0 = 2 (x) squared - 14x - 340
          0 = (x) squared - 7x - 170

Now we must solve this quadratic for x. If we use the quadratic 
formula:

   x = ( -b +- squareroot( (b) squared - 4ac ) ) / (2a)

Then a = 1, b = -7 and c = -170 and we get:

   x = ( -(-7) +- squareroot( (-7) squared - 4(1)(-170) ) ) / (2(1))
   x = ( 7 +- squareroot( 49 + 680 ) ) / 2
   x = 7/2 +- squareroot( 729 )/2
   x = 7/2 +- 27/2
   x = 34/2, x = -20/2
   x = 17, x = -10

So now you are thinking, these two values of x do not differ by 7. 
Well, they aren't supposed to. The two numbers that differ by 7 are 
x and x-7. It just turns out that there are two possible sets of 
numbers that differ by 7. Each value of x has a corresponding value 
for x-7.  The first answer is:

   x = 17 and x-7 = 17-7 = 10

And the other answer is:

   x = -10 and x-7 = -10-7 = -17

So there are two sets of numbers that differ by 7 that could be 
answers to the question. Usually the question reads "two POSITIVE 
numbers that differ by ..." and you would pick the first set, but the 
question does not exclude either set, so you can pick whichever one 
you want.

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