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Two Values of X

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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?
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```
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

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

So there are two sets of numbers that differ by 7 that could be
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/
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Associated Topics:
High School Basic Algebra

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