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For example, would solving for $x$ in $x^2=8x+7$ be the same as finding the roots of the equation? Also, would finding the roots of this be the same as finding the zeros?

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    Yes and yes. Subtract $8x-7$ from both sides and you're left with $x^2-8x-7=0$, so solutions to the former are the roots of the latter, and vice versa.2011-04-17
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    Adrian is right that "roots" and "solutions" are the same thing. Also, one finds the zeros of a _function_ by setting the function equal to zero, and then finding the roots (solutions) of that _equation_.2011-04-17
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    I think this is sloppy use of terminology. I prefer Arturo's take on things.2011-04-17
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    @Gerry: It is; sadly the terms zeroes/roots/solutions are often interchangeably used in applications.2011-04-18
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    @Gerry: what is Arturo's take on things, in this case? Or did you mean Adrian or Jesse?2011-04-18
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    @wild: Arturo deleted his answer (temporarily, I hope).2011-04-18
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    @Gerry: John Stillwell refers to [a root of the equation](http://books.google.com/books?id=V7mxZqjs5yUC&lpg=PA296&dq=root%20of%20equation&pg=PA296#v=onepage&q=root%20of%20equation&f=false) $\rm\ x^3 = 2\ $ on p. 296 of *Mathematics and its History*. A Google Books search finds similar usage of "roots of equations" by many eminent mathematicians for centuries, e.g. by Abel, Euler, van Der Waerdan, Dickson, Ore, E. Artin, Uspensky, Jacobson, Cohn, Eisenbud, Mumford, Dummit and Foote, etc. The loose terminology seems firmly entrenched - even if a bit old-fashioned.2011-04-19
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    @Bill, I take your point. Does any eminent mathematician write of solving $x^3-2$? or finding the solutions of $x^3-2$? If (as I hope) not, then there's still a distinction to be made between solving and finding roots (or zeros (or zeroes)).2011-04-19
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    @Gerry: Alas, I'll have to leave that etymological journey for someone else, since I've already invested too much time on this topic.2011-04-19
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    Just thinking out loud but could it be that a "root" (b/c square root, cube root, etc.) only applies to polynomials, while solutions/zeros could also apply to linear/trig equations?2013-11-26

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For the following pattern:

(roots of|zeros of|solutions to) (an equation|a function).

currently, the preferred usage is (in order):

  • roots of an equation.
  • zeros of a function.
  • solutions to an equation.
  • roots of a function.

enter image description here

"zeros of an equation" and "solutions to a function" are just plain wrong.

Presumably if one is talking about an equation, then 'roots' is preferred (or rather, it is just how people happen to speak).

This is all to say that these phrases are all interchangeable, but some are preferred to others in their contexts.

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    I voted this up for the imagination and effort that went into it, but I have to say we don't do mathematics by majority vote.2011-05-03
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    @Gerry: I agre with you...that mathematical content is not judged true by popularity. But mathematical language -is- (as well as mathematical fashion, what things people spend time on). Which label _must_ it be, 'normal' or 'Gaussian'? That is not judgeable by mathematics but by ...well...non-mathematical criteria.2011-05-06
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    you have a point, but I'd say that mathematical language is judged by a $\it weighted$ majority vote, where some of us have weightier votes than others. A similar chart based on usage in Inventiones might have a different look.2011-05-07