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.

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.

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!"