Negative Square Roots
Date: 01/25/2003 at 15:55:44 From: Aimee Subject: Square roots Dear Dr. Math, When you take the square root of 100 you get 10. Why can't -10 be an answer? If you square -10 you get 100 also, so why can't 10 and -10 be the answers? Sincerely, Aimee
Date: 01/25/2003 at 15:59:18 From: Doctor Ian Subject: Re: Square roots Dear Aimee, You've asked a good question. The answer depends on the context in which you're taking the square root. If you're looking for all the solutions to an equation like x^2 = 100 then you want to find _every_ value of x that satisfies the equation. There are clearly two such values: -10 and 10. However, suppose the equation arises in a context in which you're trying to find the length of the hypotenuse of a right triangle: h^2 = 6^2 + 8^2 = 36 + 64 = 100 In this context, it's clear that only the positive root is meaningful. It's not so much that h = -10 isn't a solution to the equation, but we don't know what to _do_ with a solution like that. So we ignore it. Now, suppose you turn in an exam paper with an answer like "The length of the hypotenuse of triangle ABC is either 10 cm or -10 cm". What should your teacher do with something like that? On the one hand, you've provided both solutions to the equation that you ended up with. But on the other hand, the equation was just a tool that you were using to find the length of a particular line segment. Other tools would have returned only positive values. So while -10 is a solution to the _equation_, it's not really an answer to the _question_. Do you see the difference? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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