Most people's first exposure to algebra is through simple equations like x+3=7. One thinks of x as a `mystery' number, and one solves the mystery by working out what x must be from the information given. When one is thoroughly used to this idea, one can then be taught how to solve more sophisticated kinds of equations, such as quadratic equations or simultaneous linear equations in two or three variables.
The positive integers are so basic that, when we give x=4 as a solution to the equation x+3=7, we have obviously achieved something: we have discovered that x, which we didn't know, turns out to be 4, which is a number with which we already had some familiarity. However, when we leave behind the integers, it is not always as easy to say what our achievement is. Consider, for example, the equation x2=2. What is the solution? Well, there are two, but the more obvious one is the (positive) square root of 2. And what does that mean? Well, the positive square root of two is the positive real number that squares to 2. So what have we achieved? We have `learned' that the positive solution to the equation x2=2 is the positive real number that squares to 2.
This is a technique of wide applicability. For example, the largest real solution of the quintic equation
x5-13x4+2x2 -7x-1=0
is nothing other than {[-13,0,2,-7,-1]}. And what is the meaning of the expression {[-13,0,2,-7,-1]}? Well, for real numbers a,b,c,d and e I define {[a,b,c,d,e]} to be the largest real solution of the quintic equation
x5+ax4+bx3 +cx2+dx+e=0
By a similar method I can integrate e-x2 (just define a function Phi(x) to be the integral of e-t2 from minus infinity to x), solve unpleasant partial differential equations and so on. Or, at a more basic level, I can solve the equations x+5=0 (by setting x to be -5, which is defined as the `additive inverse' of 5 - that is, the number which gives 0 when you add 5 to it) and 3x=1 (by setting x=1/3, the `reciprocal' of 3 - that is, the number which, when you multiply it by 3, gives 1).
Of course, a game that is too easy to play is an uninteresting game, and yet most of the time when we solve equations we feel that we are doing some work, and learning something that we didn't know. So what is the right way to view what we do? There are two different answers to this question, both important.
It is when we focus on rather simple equations such as x2=2, 3x=1 and x+5=0 that we are most likely to sense a certain circularity in what we do. However, in each case there is something non-trivial in what is going on. It is not a trivial statement that there is a real number that squares to 2 (as I have discussed in my pages on the square root of two and real numbers as infinite decimals ). So one way of justifying our usual practice when we `solve' the equation x2=2 is to say that, interestingly, we can prove that a unique positive real solution exists and root two is the name that we give to this solution.
This sort of answer doesn't work as well for x+5=0. It is not clear what it would mean to prove that -5 exists. However, even here there is a non-trivial statement involved, namely that the positive integers can be embedded into a larger set, which we call the integers, in which addition and multiplication can be defined in a natural way. When defined on this larger set they still have familiar properties such as commutativity, but they have the additional very useful property that every number has an additive inverse.
One can say something similar about the equation 3x=1. What is interesting about the solution is not that, amazingly, 3 has a multiplicative inverse and we have managed to find it , but rather that the ring of integers can be embedded into a field, Q.
In fact, algebraists (as opposed to analysts) take this attitude even to the square root of two. What is interesting about this number is less that it is approximately 1.4142135... and more that if we declare x to have the property that x2=2 (without worrying about what x is - though we know that it is not a rational number) then the set of all numbers of the form a+bx, with a and b rational, forms a field, which we call Q(root two). This second, more abstract, approach to equation solving is of great importance when we consider the equation x2+1=0. Again it looks circular to say that the solution is i, where i is the square root of -1, but again what is really interesting is the existence of a certain algebraic structure - this time the field of complex numbers, which of course has all sorts of interesting properties, not all of them algebraic.
The examples I have looked at so far suggest the following answer to what it means to solve an equation. The equation draws attention to an inadequacy in a certain number system (it does not contain a solution to the equation) and one is therefore driven to extend the number system by introducing, or `adjoining', a solution. With luck, the extended system has the good properties of the original one, with the added advantage that more equations can be solved.
Sometimes we may take the attitude that a larger number system has already been defined, and that to solve an equation means to prove that it has a solution in the larger number system. In other words, what is interesting is the existence (and after that, properties) of a solution rather than a neat formula for it. This switch in attitude is particularly important with differential equations - where of course now I am talking not of number systems but of systems of functions. It is known that the equation f'(x)=e-x2 has no solution in terms of functions such as polynomials exponentials and trigonometric functions. However, a solution certainly exists (even if we can't antidifferentiate, we can Riemann integrate) and it has very interesting properties, as any probabilist will tell you.
However, neither of the above views really seems to capture what is going on when we solve a quadratic equation such as x2=x+1, obtaining the solution (1 +/- root 5)/2. What do we achieve by completing the square, or by applying the formula?
The situation is undoubtedly different, because although (1 +/- root 5)/2 is the solution of the equation, this is certainly not true by definition . On the other hand, part of the solution, namely the square root of 5, is a by-definition solution. So it seems that what we have done is to take for granted that we can solve the equation x2=5 (and similar ones) and to use that interesting ability to solve an equation which is not of such a simple form.
In other words, when we `solve' the quadratic, what we are really doing is showing that the problem can be reduced to that of solving a particularly simple quadratic - that is, one of the form x2=c. A similar remark can be made about the equation 3x=7. Suppose we know that multiplication and addition satisfy all the axioms for a field. That allows us to solve, by definition, an equation like 3x=1, but to solve the equation 3x=7 we must take the additional step of multiplying the multiplicative inverse of 3 by 7. Thus, from the existence of multiplicative inverses we can deduce the solutions of all equations of the form ax=b (with a not equal to zero).
Similarly, one might wish to say that ex is by definition the solution of the differential equation f'(x)=f(x) (with f(0)=1). Once that is done it is interesting to know that no new definitions are needed in order to solve equations such as f''(x)-3f'(x)+2f(x)=0.
This point of view is the one taken when one says that quintic equations cannot be solved. It can be shown that solutions exist (as real or complex numbers, or as abstract objects adjoined to the rationals) but it is not possible to reduce the solution of the quintic to the solution of equations of the form xm=c. To be a little more precise, there is no formula for the solution of a general quintic, where the formula expresses a function of the coefficients and the function is a composition of the usual arithmetical operations together with the extra operation of taking roots.