"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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**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!"