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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. – Adrian Petrescu Apr 17 '11 at 22:58
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. – Jesse Madnick Apr 17 '11 at 23:00
I think this is sloppy use of terminology. I prefer Arturo's take on things. – Gerry Myerson Apr 17 '11 at 23:53
@Gerry: John Stillwell refers to a root of the equation $\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. – Bill Dubuque Apr 19 '11 at 2:38
@Gerry: Alas, I'll have to leave that etymological journey for someone else, since I've already invested too much time on this topic. – Bill Dubuque Apr 19 '11 at 3:42

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. – Gerry Myerson May 3 '11 at 0:17
@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. – Mitch May 6 '11 at 14:00
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. – Gerry Myerson May 7 '11 at 4:10

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