Practice problems
First of all, consider these two similar, but not identical, QCs.
All I will say right now is: despite apparent similarities, those two questions have two completely different answers. The distinction between them is the subject of this article.
Do I include the negative root?
Often, students are confused about this question. For example, in the questions above, we know +4 is a square root of 16, but isn’t -4 as well? Do we include it as part of Column B or not? Does it matter how the question is framed? All of these questions are resolved by understanding the following two cases.
Case I
In case one, the test-maker, in writing the question, uses this symbol. This symbol appears printed on the page as part-and-parcel of the question itself.
What is this symbol? Well, the benighted masses call this simply a “square-root” symbol, but the proper name is the “principal square root” symbol. Here, “principal” (in the sense of “main” or “most important”) means: you take one and only one root, the most important one —i.e. the positive root only. That is the deep meaning of this symbol.
Thus, in this case, cases in which the test maker, by the sheer process of writing the question itself, has written down this symbol, and this symbol appears as part-and-parcel of the question itself, then you NEVER consider the negative root, and ONLY take the positive root.
Case II
In this case, that special symbol does not appear as part of the problem. What does appear is, possibly, a variable squared, or some other combination of algebra that leads to a variable squared, and you yourself, in your process of solving the problem, have to take the square-root of something in order to solve it. The act of “square rooting” is not initiated by the test maker, in the very act of writing the question, but is left to you, the problem solver, to initiate.
In this case, 100% of the time, you ALWAYS have to consider both the positive and negative square roots.
Summary
If you master that distinction, you will always understand when to consider both positive and negative roots, and when you need only consider the positive root. You may want to go back to those two QCs at the beginning and think them through again before reading the solutions below.
Practice problem solutions
1) Here, the principal square root symbol appears printed as part of the problem itself. We are in Case I. Of course, that symbol implies: take the positive square root only. So Column B can only equal +4. Of course, that’s always bigger than 3, so Answer = B.
2) Here, there’s no square root symbol printed as part of the problem itself. We are in Case II. For any square roots we take as part of our solution, we are liable to account for both the positive and negative roots.
Sure enough, the very first thing we encounter in the prompt is a variable squared, and when we solve for x, we have to account for both roots: x = ±4. The variable x could equal well have either one of those values.
Now, when we proceed to the QC, we see that the different values of x would give different answers. If x = +4, then column B is greater, but if x = -4, then column A is greater. Different values lead to different conclusions, and this situation means we don’t have enough information to establish a definitive relationship. Answer = D
I was benighted of this trick !! Thnxx a lot Mike
You are quite welcome, my friend.
Mike
Thanks for making the distinction (I would have missed that). I just became a member today and I already find the lessons and explanations invaluable. I feel like I’m stealing from you guys (don’t get any ideas). All kidding aside, thanks for doing what you do because I was going the Kaplan route, but the Kaplan route is not for me.
Congratulations on joining Magoosh: I think you will see tremendous benefit in your score. Thank you very much for your kind words.
Mike
Thanks Mike ! I was always confused with such questions. This post makes things a lot clearer !
You are quite welcome.
Mike
Wow! Thanks for these blog posts. The more I read, the more I am amazed at my own ignorance. I also start to feel a bit lost in the sea of things that I don’t know.
Have faith, my friend. You can master this stuff! Thank you for your kind words.
Mike
Thanks Mike — Really helpful.
You are very welcome.
Mike
Hi Mike, it’s great that you pointed out that the radical sign √ refers to the principal (positive) square root only. (many people use it without realizing that)
In the quadratic equation, ± is used in front of √ (b2-4AC) so that the negative root is included in the equation…..and not just the positive root.
source: square root calculator
Anthony:
Exactly: the √ sign means “positive only”, so in any context in which both the ± roots are required (as in the Quadratic Formula), we need to add the ± sign in front of the √ . What you have shared is 100% correct. Thank you.
Mike
Hi Mike, Many thanks for pointing out this important difference . It seemed innocuous , and most people (including me) would have failed to realize this fine distinction.
Thanks to you, now I’ll be on a lookout
Regards,
Anupam
You are quite welcome. Thank you for your kind words.
Mike