I needed to review the quadratic formula before addressing it in class, mostly to make sure I could work my way through it smoothly, but partially I must admit to be sure I could do it at all.
Turns out I can. No sweat. Compared to some of the gyrations of differential equations or network optimization, the math is straightforward as long as you grasp the method in the first place. Again, compared to more advanced math, the formula itself looks pretty simple.
But what a frankensteinian monster it appears to students still getting used to algebra! In order to isolate x in a quadratic, you must reduce a quadratic polynomial to a linear one. To do this, you must employ a square root, since dividing by x just leaves you with a troublesome 1/x floating about somewhere else. In order to employ a square root, you must have a proper square to work with. And since a general quadratic isn’t a proper square, you have to build one from spare parts. You do this by hacking off any bits that are too big to fit (the c), sewing on new bits (b squared over 4a) to fill in the missing quantities—quantities that seem arbitrary until you’ve peeked ahead at the answer—taking the square root of all the junk that’s accumulated in balancing the equation, subtracting, dividing, simplifying, and running electricity through the whole mess until it rises to terrorize the peasantry. Small wonder the final formula is so messy.
Not difficult, if you know where you’re going and keep a firm grip on why you’re going through all these operations: to reduce the quadratic to a linear expression with a square root. But arcane, even ridiculous, if you don’t really know where you’re going and why, which is all too often the case with math students.
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