What (else) it means to solve

In class yesterday, I had an opportunity to reinforce what it means to solve equations and inequalities (a connection I've discussed in a previous post). My students and I were working on a task that required that we verify that x = 2 and x = 0 are the two solutions of the quadratic equation -2x^2 + 4x + 1 = 1. How can we do that? A chorus of replies: "plug them in and check." 

"Right," I said, "We need to see if we have found both (all) of the values that make the equation true."

Learner #1 raised his hand and asked, "Doesn't it also mean finding the points where the parabola intersects the line y = 1?" I know from a previous conversation (described in this previous post) that this was how he learned to solve quadratics in high school: enter the parabola into Y1, enter the other expression into Y2, and use the CALC -> INTERSECT to solve.

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So I was happy to affirm the connection.

A short time later, in the same lesson, we were exploring the function f(x) = (x-3)^2 + 5. We had determined that because the parabola opened upward and had its vertex at (3,5), so there must not be any solutions to the quadratic equation f(x) = 1. Furthermore, we agreed, the equation f(x) = 5 would have just one solution (at the vertex) and f(x) = k for any k > 5 would have two real solutions.

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Then Learner #2 offered: "So if we ever find that there's no solution, we can conclude that the curves do not touch, right?" It was a nice extension of Learner #1's remark.

But it depends a bit on what you mean by "no solution," doesn't it? The situation is exactly analogous to the 1st grader who asks the question, "Teacher, what's 1 minus 3?" to which the teacher response helpfully, "Honey, you can't do 1 minus 3." I like to imagine a precocious 1st grader reply: "Yeah, maybe you can't, but I can! It's negative 2!"



Of course, every subtraction problem has a solution, although you may need to use negative numbers to express it. Similarly, every quadratic equation has two solutions (a consequence of the fundamental theorem of algebra), although you may need to use complex numbers to express them.

And according to Student #1's observation, if the equation has two solutions, the graphs must intersect in two places!  (...But where?)

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A picture formed in my mind of an invisible, ethereal, wormhole-style thread binding the two curves. How could we represent the functions so that the "intersection" would be visible?

In teaching, decisions are made quickly, often based on little more than intuition about the most fruitful direction in which to proceed. Decision time: how should I respond to Learner #2's observation?

What would you have said??

I said: "Exactly right, except... remember when we said that every quadratic equation has two complex-valued solutions? That means that these curves that appear NOT to intersect really DO have two points in common." I paused for a moment, then added, "Isn't that crazy?! Why can't we see them?!"

I noticed a few crooked smiles emerge as ideas began to percolate. I made a vague reference to the bizarre but powerful realm of complex numbers, adding that students who moved on to take the next course in the sequence would get to explore them a bit, and then we returned to our originally scheduled programming.

But I came back to thinking about it after class. I couldn't help but wonder if there was a way to get that invisible wormhole linking the y = f(x) and y = 5 to reveal itself. In my next post, I'll explore the issue of visualizing the solutions to quadratic functions like these.

My follow-up post: Finding wormholes
My initial post in this series: What it means to solve