{"id":11133,"date":"2016-09-07T15:52:50","date_gmt":"2016-09-07T22:52:50","guid":{"rendered":"https:\/\/magoosh.com\/gre\/?p=11133"},"modified":"2019-01-09T19:59:20","modified_gmt":"2019-01-10T03:59:20","slug":"positive-and-negative-square-roots-on-the-gre","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gre\/positive-and-negative-square-roots-on-the-gre\/","title":{"rendered":"Positive and Negative Square Roots on the GRE"},"content":{"rendered":"

Some questions to start: Can a square root be negative?<\/em> Yes, there is such a thing as a negative square root. How would you deal with negative roots on the GRE?<\/em> This question is actually fairly complicated, because it depends on how the GRE words the question. Let’s start with some GRE math<\/a> practice problems before I explain.<\/p>\n

\"negative<\/p>\n

Negative square root practice problems<\/h2>\n

First of all, consider these two similar, but not identical, Quantitative Comparison questions.\u00a0 These are a bit easier than you would see on the GRE, but they illustrate an important distinction about possible negative square roots.<\/p>\n

1)<\/p>\n

\"QC<\/p>\n

2)<\/p>\n

\"QC<\/p>\n

All I will say right now is: despite apparent similarities, those two questions about whether roots are positive or negative, have two completely different answers<\/em><\/strong>.\u00a0 The distinction between them is the subject of this article.<\/p>\n

Explanations to these practice problems will appear at the end of this blog article. Jump ahead by clicking here<\/a>.<\/p>\n

Do I include the negative roots?<\/h2>\n

Often, students are confused about this question.\u00a0 For example, in the questions above, we know +4 is a square root of 16, but can’t -4 be one as well? Or can it? \u00a0Can a square root be negative?\u00a0 Do we include the negative square root as part of Column B or not?\u00a0 Does it matter how the question is framed?\u00a0 All of these questions about possible negative square roots are resolved by understanding the following two cases.<\/p>\n

Case I: the symbol appears<\/h2>\n

In case one, the test-maker, in writing the question, uses this symbol.\u00a0 This symbol appears printed on the page\u00a0in\u00a0the question itself.<\/em><\/p>\n

\"pnsr_img2\"<\/p>\n

What is this symbol?\u00a0 Well, the most folks call this simply a “square-root” symbol, but the proper name is the “principal square root” symbol.\u00a0 Here, “principal” (in the sense of “main” or “most important”) means: you take one and only one root, the most important, or principal, one \u2014 the\u00a0positive root only<\/a>.\u00a0 That is the deep meaning of this symbol.<\/p>\n

Thus,\u00a0in all cases in which\u00a0this symbol appears as part\u00a0of the question itself, you NEVER consider the negative square root, and ONLY take the positive square root.<\/p>\n

Case II: the symbol doesn’t appear<\/h2>\n

In this case, that special symbol does\u00a0not<\/strong>\u00a0appear as\u00a0part<\/em>\u00a0of the problem.\u00a0 What does appear is, for instance, a variable squared, or some other combination of algebra that leads to a variable squared, and you yourself, in your\u00a0process<\/em>\u00a0of solving the problem, have to take the square root of something in order to solve it.\u00a0 The act of “square rooting” is not initiated by the test maker in the act of writing the question; rather, it is you who initiated the square-rooting.<\/p>\n

In this case, 100% of the time, you ALWAYS have to consider\u00a0both<\/strong>\u00a0the positive and negative square roots.<\/p>\n

Summary: Can a square root be negative?<\/h2>\n

Can a square root be negative?\u00a0 Well, the answer is: it depends on what was printed in the problem.\u00a0 The principal square root symbol never has a negative output, so if the test maker printed that symbol, it’s restrictions have to be respected: all square roots then are positive.\u00a0 On the other hand, if the problem contains a variable squared, or some other algebra that leads to a variable squared, and you yourself take a square root as part of the act of solving, then you always have to consider all possible solutions, both the positive square roots and the negative square roots.<\/p>\n

If you master that distinction, you will always understand when to consider both positive and negative roots vs. when to consider only the positive root.\u00a0 You may want to go back to those two QCs at the beginning and think them through again before reading the solutions below.<\/p>\n

Additional resources<\/h2>\n

If you’re having trouble with Quantitative Comparison questions on the GRE, I would recommend taking a look at some of these additional resources. They may help clear up some of the concepts that you’re struggling with, and can provide some extra practice.<\/p>\n

Quantitative Comparison Tips<\/h3>\n