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It is known that, in the universe of complex numbers, the only root of the equation $x^2 - 2x + 1 = 0$ is $1$. Could we say that the equation has two equal real roots? Or should we say that the equation has one real root with multiplicity 2?

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I was under the impression that those are just two ways of saying the same thing. –  crasic Nov 15 '10 at 10:41
    
There's a reason why the statement of the Fundamental theorem of Algebra has the phrase "counting multiplicity" or something equivalent in its statement. One also hears the word "coalesce" in some contexts. –  J. M. Nov 15 '10 at 10:50
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4 Answers 4

"Two equal roots" and "a root of multiplicity (at least) two" mean the same thing.

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Why at least? Does it mean that two equal roots and a root of multiplicity 3 (or 4, 5 etc) mean the same thing? –  Dilawar May 14 '12 at 7:02
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@Dilawar: Because language is inexact; if I had a root of multiplicity 3, we would have "three equal roots", but if you have three things that are equal, then you also have two things that are equal. Of course, with a quadratic, multiplicity cannot exceed 2 in the first place. –  Arturo Magidin May 14 '12 at 15:18
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If you want to be totally precise, you have to say that it has one real root with multiplicity 2. But "two equal real roots" is almost universally understood as a shortcut for saying that.

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Both are correct, as well as saying that the root is degenerate (with multiplicity of two).

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Yes, all these answers are correct. I am not allowed to upvote them since I'm not registered, but I do know that the parabola(or quadratic function) you have there touches the graph on the x-axis in precisely 1 point. That is the reason why it is the solutions to your quadratic will be a rational, real, and equal root.

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Consider registering for an account so you can upvote! :) –  J. M. May 14 '12 at 7:02
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