Square Roots and False Solutions

Date: 22 May 1995 12:08:36 -0400
From: Anonymous
Newsgroups: local.dr-math
Subject: Squaring Variables

Why does squaring a variable sometimes create a false solution?
RLM

Date: 22 May 1995 13:39:43 -0400
From: Dr. Ken
Newsgroups: local.dr-math
Subject: Squaring Variables

Hello there!

Here's why.  When you do something to both sides of an equation, what you're
really saying is this: if I have two things that are equal, and I do the
same thing to both of them, what I get should be two things that are equal.
For example, if you know that x-6 and 8 are the same thing, i.e. x-6=8, and
you add 6 to both objects, i.e. make the equation x=14, you're guaranteed
that the new equation is true.

Now let's look at the case where you're squaring both sides of an equation.
If we know that x and 12 are the same thing, we can square both of them and
get x^2 = 144.  That's all fine; x squared really does equal 144.  But
notice that if we wanted to use this new equation to find x, we'd have to
take the square root of both sides.  Here's where the problem is: every
positive number has TWO square roots, one the negative of the other!  So if
we took the square root of both sides, we wouldn't get just x=12, we'd also
get x=-12, -x=12, and -x=-12 (these four equations come from the two square
roots of each side: x, -x, 12, and -12).  Note that these four equations are
the same thing as x=12, x=-12.  So the extra solution happens because every
positive number has two square roots instead of just one.

Thanks for the question!

-K