Algebra I Extra Credit Problems, Q#1

[caa8q]

Hi all,

I'm sorry for the elementary nature of this question related to algebra rules, but I am hoping you can help. The question is as follows:

If ((a+b)^2) / ((a-b)^2) = 1, then which of the following statements must be true?

I. a=0
II. b=0
III. a= -b

(A) None
(B) I only
(C) II only
(D) III only
(E) I, II, and III

The answer is (A). The answer does a good job of explaining how to get the right answer: set the denominator and the numerator equal to each other, expand each polynomial, and then get variables on both sides of the '=' to cancel out. That leaves us with ab=0. So, (A) is the answer.

However, I did this problem differently and got it wrong. I know that my method is wrong, but could someone please explain to me why, and where I went wrong? In particular, what is wrong with NOT expanding the polynomials? This was my process:

(a+b)^2 = (a-b)^2
sqrt ((a+b)^2) = sqrt((a-b)^2)
a+b = a-b
2b=0
b=0

Thank you for your help

[RonPurewal]

the problem there is that "this^2 = that^2" doesn't necessarily imply that this = that; it's just as possible that this = -that.
for instance, if this = -3 and that = 3, then "this^2 = that^2" is still a perfectly true statement.

the same thing is true here: if (a + b)^2 = (a - b)^2, then it's possible that a + b = a - b... but it's just as possible that (a + b) and (a - b) are opposites.
if you set up the equation in which they're opposites (i.e., a + b = -(a - b), or -(a + b) = a - b], then you'll get a = 0, rather than b = 0.

--

if you still have conceptual trouble with the above examples (yeah there are a lot of a's and b's flying around up there), just consider a really simple example.
say i tell you x^2 = 4... can you "square root both sides" and just tell me that x = 2?
nope; x can also be -2.
same problem here.

[RonPurewal]

oh, and, don't forget, you can also resolve this problem by just plugging in numbers, even if you don't have clue number one how the algebra works.

in particular, you know you're interested in the possibilities "a = 0" and "b = 0", since, well, those are the things that are there. (in fact, more precisely, you're interested in whether these statements can be false, since that would immediately give a "no" to "must be true?".)

so, just try some really simple possibilities in which these things are true or not true.

* the first pair i'd personally think to try would be a = 0 and b = 0. but, a quick glance at I/II/III shows me that it'd be pointless to try those, since they satisfy all three statements (and thus can't eliminate anything).

* try a = 0, b = 1.
... then the equation becomes 1^2/(-1)^2 = 1. that's a true statement.
b is not 0, so II is eliminated.
a and b are not opposites, so III is gone too.

* try a = 1, b = 0.
... then the equation becomes 1^2/1^2 = 1. that's also a true statement.
a is not 0, so I is eliminated.

done.

[caa8q]

Thank you very much for your thorough explanation

[RonPurewal]

Thank you very much for your thorough explanation

sure.