Square Roots and False SolutionsDate: 22 May 1995 12:08:36 -0400 From: Anonymous Newsgroups: local.dr-math Subject: Squaring Variables Why does squaring a variable sometimes create a false solution? RLM Date: 22 May 1995 13:39:43 -0400 From: Dr. Ken Newsgroups: local.dr-math Subject: Squaring Variables Hello there! Here's why. When you do something to both sides of an equation, what you're really saying is this: if I have two things that are equal, and I do the same thing to both of them, what I get should be two things that are equal. For example, if you know that x-6 and 8 are the same thing, i.e. x-6=8, and you add 6 to both objects, i.e. make the equation x=14, you're guaranteed that the new equation is true. Now let's look at the case where you're squaring both sides of an equation. If we know that x and 12 are the same thing, we can square both of them and get x^2 = 144. That's all fine; x squared really does equal 144. But notice that if we wanted to use this new equation to find x, we'd have to take the square root of both sides. Here's where the problem is: every positive number has TWO square roots, one the negative of the other! So if we took the square root of both sides, we wouldn't get just x=12, we'd also get x=-12, -x=12, and -x=-12 (these four equations come from the two square roots of each side: x, -x, 12, and -12). Note that these four equations are the same thing as x=12, x=-12. So the extra solution happens because every positive number has two square roots instead of just one. Thanks for the question! -K |
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