Extraneous RootsDate: 06/13/2001 at 00:48:26 From: Kostya Tomashevsky Subject: An equation with a square root Recently I came across this equation: sqrt(x-3) = x-5 I squared both sides to solve the quadratic equation, and got the solutions 4 and 7. When I checked them I saw that 4 only worked when I calculated the square root as a negative, and 7 only worked when the root was assumed positive. My question is, are both of these solutions valid, or only 7? Or neither of them? Thank you, Kostya Tomashevsky Date: 06/13/2001 at 08:41:23 From: Doctor Peterson Subject: Re: An equation with a square root Hi, Kostya. What you have discovered is called "extraneous roots." You can read about them here: Why Multiple Roots? http://mathforum.org/dr.math/problems/duff.9.9.96.html What happens is that, by squaring both sides of the equation, you have made an equation whose roots include those of both sqrt(x-3) = x-5 and -sqrt(x-3) = x-5 because these two equations give the same result when you square them. If you treated the radical in the original problem as having two values, positive and negative, then both of the roots you found would be valid; but since we interpret a radical as a function that returns only the positive root, only one works. You can't be sure which root will be valid for the original equation until you check them. You always have to check each solution you find, and in this situation only those that work are valid solutions to the problem. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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