![]() The variable x could equal well have either one of those values. 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. For any square roots we take as part of our solution, we are liable to account for both the positive and negative roots. Answer = (B).Ģ) Here, there’s no square root symbol printed as part of the problem itself. Of course, that symbol means: take the positive square root only. GRE Math: Solving Quantitative Comparisonsġ) Here, the principal square root symbol appears as part of the problem itself.QC: “The relationship cannot be determined from the information given” Answer Choice. ![]() They may help clear up some of the concepts that you’re struggling with, and can provide some extra practice. ![]() If you’re having trouble with Quantitative Comparison questions on the GRE, I would recommend taking a look at some of these additional resources. You may want to go back to those two QCs at the beginning and think them through again before reading the solutions below. If you master that distinction, you will always understand when to consider both positive and negative roots vs. 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. 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. Summary: Can a square root be negative?Ĭan a square root be negative? Well, the answer is: it depends on what was printed in the problem. In this case, 100% of the time, you ALWAYS have to consider both the positive and negative square roots. 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. 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 process of solving the problem, have to take the square root of something in order to solve it. In this case, that special symbol does not appear as part of the problem. Thus, in all cases in which this symbol appears as part of the question itself, you NEVER consider the negative square root, and ONLY take the positive square root. Here, “principal” (in the sense of “main” or “most important”) means: you take one and only one root, the most important, or principal, one - the positive root only. What is this symbol? Well, the most folks call this simply a “square-root” symbol, but the proper name is the “principal square root” symbol. This symbol appears printed on the page in the question itself. In case one, the test-maker, in writing the question, uses this symbol. 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? Can a square root be negative? Do we include the negative square root as part of Column B or not? Does it matter how the question is framed? All of these questions about possible negative square roots are resolved by understanding the following two cases. Often, students are confused about this question. The distinction between them is the subject of this article.Įxplanations to these practice problems will appear at the end of this blog article. ![]() The irrational numbers together with the rational numbers constitutes the real numbers.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. The decimal form of an irrational number will neither terminate nor repeat. This means that they can't be written as the quotient of two integers. The square roots of whole numbers that are not a perfect square are members of the irrational numbers. In the first section of Algebra 1 we learned that
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