Help Victoria! (or anyone else)

You’re not delusional, at least not in this area, although I can’t say about the rest of your life …You’ve been caught by ** New Math Standard Trick Question Number 5b.**The sign problem: when you square 3, 3 x 3 = 9. When you square -3, (-3) x (-3) = +9 also. So, +9 has two real square roots, either +3 or -3. When doing a real-world problem, you have to look at *both* possible answers and see if one or the other or maybe either fits the requirements of your problem. So far so good. But this doesn’t sit well with new math type people, who can’t deal with ambiguity. Actually, there is one good reason not to have ambiguous answers, and that is because computers and calculators can’t deal with ambiguity either.So we set up what is called a *convention*. A convention is an arbitrary rule that we make up so everybody knows where the lines are. There’s no logical reason to choose one convention over another (That’s why it’s a convention, not a mathematical law), but we have to pick one to avoid confusion. An example of a real-world convention is which side of the road to drive on. England, Trinidad, and I believe still Japan (used to, anyhow) drive on the left side of the road. North America, Europe, and most of the rest of Asia drive on the right. Sweden used to drive on the left but switched to the right in the 1960’s when thay started to sell a lot of Volvos overseas. Now, either system works fine. England has safe roads. But if you go to England, or if a British person comes here, you have to remember forcefully to change sides of the road to avoid head-on collisions. Any convention works as long as everybody you come into contact uses the *same* convention. Another convention is the direction of reading; English and European left-to-right, Arabic and Hebrew right-to left. Both work, just remember which situation uses which.OK, the mathematical sign convention as formalized in the 1960’s:(1) Numbers showing signs, eg +3, -4, are as written **(2) Numbers *without* signs are ALWAYS taken to be positive, eg 4 = +4 (3) Letters can stand for numbers of any sign, or zero (4) Putting a negative sign in front of a letter, eg -b, means “take the opposite”. If b is positive, -b is negative; and if b is negative, -b is positive ( Think a second; -(-5) = +5 )So, the square root of a number is ALWAYS assumed to be the *positive* choice of square root, unless noted differently. (because it has no sign and falls under rule 2)Using the symbol to mean “square root”, 9 = +3 by convention.If I mean that I can take either square root (as I *must* in solving quadratics) I have to use the “plus or minus” symbol. We write this as a plus over a minus, but I’ll type it as +-.We also type x squared as x^2So given the problem x^2 = 9, the solution is x = +- 9, or x = +-3 (Still using for root)Note that by this new convention, if I mean both square roots are possible solutions to the problem (as they are), I have to state explicitly when I write the root that I can take both of them, by using the +- sign.Now, back to your problem.Suppose x = 3. Then x^2 = 9. Then (x^2) = 3, and yes, (x^2) = x in this case. (again using for square root sign)Now suppose x = -3. The x^2 = +9. Then (x^2) = +3 by the sign convention. Oops, our sign shifted! Here, (x^2) = the *opposite* of our original x. Using the other sign convention about opposites, -(-3) = +3. So here, (x^2) = -xSo you have a headache yet? Are you climbing the walls saying what a bunch of nitpicking #$%&^&***( ??Good, you have some common sense and a sense of priorities.This kind of stuff is of minimal importance to beginning algebra and really isn’t relevant in Algebra 1. Later, in Algebra 2, a small amount of time should be spent on it, and people who really get into advanced math have to be careful of these details and should spend some time clearing them up in university. These details do matter in making sure that the solution you find really is applicable to the real world, but not in first year studies.Anyway, the answer to your question is:(x^2) = x? Sometimes, if x is zero or positive.(x^2) = -x? Sometimes, if x is zero or negative.Note that zero works both ways, since -0 = +0A summary statement (actually a useful one!) is(x^2) = |x| where |x| is the **absolute value** of x: I’ve just been reviewing my son’s most recent algebra test. There are
: two questions he left blank that I’m pretty sure I know the answer
: to but I’m not 100% positive. I want to make sure he understands
: the concept so I’m not about to make any guesses and pass them off
: as fact: Unfortunately I can’t type the square root or exponent symbols so I
: have to translate into words.: OK here goes: Answer the following with Sometimes, Always or Never.: 1.) the square root of X squared = X: 2.) the square root of X squared = -X: Having not taken math since dinosaurs roamed and not recalling ever
: being asked these questions when I was in school…: My intuition tells me that although the square root of 9 can be 3 or
: -3, the square root of X squared would always X. Of course the
: variable X may stand for either a positive or negative number. If
: I wanted to denote the square of negative X I would use the
: negative sign to the left of X squared.: Using this logic my answers to the questions would be: 1.) Always 2.)
: Never: Am I correct or dillusional? Is my reasoning correct or am I missing
: something?

The answer is sometimes
The square route of x squared is either +x or -x
Thus it is sometimes +x and sometimes -x
If you were to say it is always +x then you are automatically excluding the fact that it is also -x
It would perhaps be more correct to say that x squared always has two roots, one positive and the other negative.
I hope this is helpful
Ann

Hi, Ann. This is one of those cases where the teacher and/or textbook is trying to catch the student out. If you square -3, the result is +9. But if you take the square root of 9, the answer is +3 because these books insist on removing all ambiguity, so you have to pick one and only one answer and the conventional choice is the positive root. They no longer allow you to say you have two roots. So as you said the square root of x squared is *either* x *or* -x (but they won’t accept both) depending on the original sign of x. What they really want to say — but lose the meaning in the details — is that the square root of x squared is the absolute value of x. The kid (and parent) was quite understandably confused by all this detail.