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Algebra 2

``I'm thinking of a number, and its name is `x' ...'' So if

a x² + b x + c = 0, (4.14)

what is x? Well, we can only say, ``It depends.'' Namely, it depends on the values of a, b and c, whatever they are. Let's suppose the dimensions of all these ``parameters'' are mutually consistent^4.7 so that the equation makes sense. Then ``it can be shown'' (a classic phrase if there ever was one!) that the ``answer'' is generally^4.8

$$x = {-b \pm \sqrt{b^2 - 4ac} \over 2a} .$$ (4.15)

This formula (and the preceding equation that defines what we mean by a, b and c) is known as the Quadratic Theorem, so called because it offers ``the answer'' to any quadratic equation (i.e. one containing powers of x up to and including x<sup>2</sup>). The power of such a general solution is prodigious. (Work out a few examples!) It also introduces an interesting new way of looking at the relationship between x and the parameters a, b and c that determine its value(s). Having x all by itself on one side of the equation and no x's anywhere on the other side is what we call a ``solution'' in Algebra. Let's make a simpler version of this sort of equation:

``I'm thinking of a number, and its name is `y' ...'' So if y = x<sup>2</sup>, what is y? The answer is again, ``It depends!'' (In this case, upon the value of x.) And that leads us into a new subject....