Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

Perfect Square: Solving Two Equations


Date: 6/21/96 at 20:58:1
From: Sheldon Wilson
Subject: Find x so that Two Expressions are Perfect Squares

Find the value of x such that

        x^2 + 5 is a perfect square

        AND

        x^2 - 5 is also a perfect square.

The value of x must not contain surds; that is,
x must equal M / N where M = {1 or 2 or 3 or 4 or . . .} 
and N = {1 or 2 or 3 or . . .}

I have solved this problem correctly but I am not satisfied 
with my solution. I am looking for a more elegant solution. 
Thanks. 


Date: 6/26/96 at 11:5:38
From: Doctor Brian
Subject: Re: Find x so that Two Expressions are Perfect 
Squares

Try looking at some of the differences between perfect 
squares. You're trying to find a difference of ten (between 
x^2 - 5 and x^2 + 5)

1-0=1
4-0=4   4-1=3
9-0=9   9-1=8   9-4=5
16-0=16 16-1=15 16-4=12 16-9=7
25-0=25 25-1=24 25-4=21 25-9=16 25-16=9

At this point, the difference between *consecutive* perfect 
squares is more than ten:
(x + 1)^2 - x^2 = x^2 + 2x + 1 - x^2 = 2x + 1, which for x>5 
gives an answer greater than or equal to 11.

Finally, since squaring a number is an increasing function, 
then any larger gap between the numbers must give a larger 
difference that between consecutive numbers.  For instance, 
the gap between the squares of 6 and 8 must be larger than the 
difference in the squares of 6 and 7.

So the only possible way to give a difference less than 10 
between the numbers would be to use numbers less than five.  
But the above table shows that there is no pair of numbers 
less than five whose squares give a difference of ten.

-Doctor Brian,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Basic Algebra

Search the Dr. Math Library:


Find items containing (put spaces between keywords):
 
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994-2013 The Math Forum
http://mathforum.org/dr.math/