Q. "How do I explain to my 15 year old son the value of knowing the quadratic equation in everyday life?"

A. In part the answer depends on your definition of "everyday life." If by the mathematical activities of everyday life you restrict yourself to balancing your checkbook or figuring out how much you'll owe altogether in purchasing a few objects whose prices you know individually (or perhaps figuring out their average price), then you may never need to solve a quadratic equation. Up to this point I probably agree with your son.

But the world is an interesting and complicated place, and our attempts to understand it often require us to solve equations that describe relationships among different quantities, and this in turn may require the quadratic formula.

I'll give an example involving area. Suppose we want to make a strong cardboard box for shipping baseball bats. The box should have a square base and be 3 feet high. The box manufacturer's machinery will use 25 square feet of cardboard to make each box. Here's the question: Given all this information, what should the dimensions of the box be?

We already know that the height of the box is to be 3 feet. The top and bottom are squares, and what we don't know is the side length of those squares. Let's call that unknown value x. So the top and bottom will each have area x^2. (Here I use "^" to represent exponentiation. So x^2 represents the square of x.) Each of the four sides of the box will have area 3x, since the box is to be 3 feet tall. So the total surface area (in square feet) of the box---gotten by adding the areas of the top, bottom, and four sides---will be 2x^2 + 12x; and since the manufacturer will use 25 square feet of cardboard to make the box, we must have 2x^2 + 12x = 25; that is, our unknown value x must satisfy the quadratic equation 2x^2+12x-25=0.

With the quadratic formula in hand, this equation can be solved for the value of x.  I'll let your son do that.  When he does, he will find that there are two values for x that satisfy the equation: one negative and one positive. Upon seeing a negative value for x, his first thought may be "This is ridiculous, because no one ever heard of a box with a negative dimension!"  Here it must be remembered that the equation doesn't "know" that the problem involves boxes or dimensions. We need to think to ourselves that while we've shown that the value of the unknown length x must satisfy the equation, other numbers too---having nothing to do with our original problem---may also satisfy the equation.  So in the present situation we will discard the negative answer, and the positive answer is the desired edge length of the square base.

Now the objection may come that "areas are boring." But it turns out that areas are fundamental for the development of calculus, and calculus in turn can be used to solve an extraordinary range of problems having no obvious connection to areas at all! So the quadratic formula is a fundamental tool for applying mathematics to our description of the universe.

I hope this answers your son's question.

Larry Gerstein is a professor in the Department of Mathematics at UC Santa Barbara. He specializes in quadratic forms and number theory.

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