Perfect Squares and Inspection

Date: 23 May 1995 08:31:25 -0400
From: MUHTAR
Newsgroups: local.dr-math
Subject:square roots

2+2=?,yapamazsaniz bana sorun !!!!
x^1/2+y=11
y^1/2+x=7
what are X?,Y?

From: Dr. Ken
Newsgroups: local.dr-math
Subject: Re: square root problem
Date: 23 May 1995 10:41:46 -0400

Hello there!

>2+2=?,yapamazsaniz bana sorun !!!!

What?

I think this is an example of a problem that is best solved the way that
most school teachers tell you not to solve math problems: by inspection.  The
first thing you should notice about the problem is that you've got square
roots on the left side, but integers on the right side.  Now recall the fact
that the only way that the square root of a number can be rational is if the
number is a perfect square (that's kind of the definition of a perfect
square).  So it's possible that x^1/2 is irrational and adding y to it
gives you an integer, but it's more likely that the two numbers are perfect
squares.  Let's go with this strategy first.

We know that both numbers are going to be positive, because if one were
negative we'd get imaginaries in the left side.  So looking at the first
equation, what are the perfect squares that are less than 11?  These will be
your values for y.  What does each of these values make x?  Try all of these
values in the second equation, and you should have your solution!  

-K