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?
Is solving an equation the same as finding the roots?
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algebra-precalculus
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0Yes 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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2Adrian 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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3I think this is sloppy use of terminology. I prefer Arturo's take on things. – 2011-04-17
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0@Gerry: It is; sadly the terms zeroes/roots/solutions are often interchangeably used in applications. – 2011-04-18
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0@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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0@wild: Arturo deleted his answer (temporarily, I hope). – 2011-04-18
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1@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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0@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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1@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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0Just 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