Question review: the equation with a constant in it, and the Desmos move that replaces the discriminant

Second in the series: one question, worked more than one way, with the clock running. This one is the question type that convinced me the Desmos calculator is a different tool from the one students grew up with, and it has a trap that catches students who already use Desmos a lot.

The question

The equation 3x² − 36x + k = 0 has exactly two real solutions, where k is a constant. If k < m, what is the least possible value of m?

A box to type into. Advanced Math, usually late in the second module, and it has a family of cousins: exactly one real solution, no real solutions, for what value of k, for which values of c. The method is the same for all of them.

By hand

A quadratic ax² + bx + c = 0 has two real solutions when its discriminant, b² − 4ac, is positive. Here a = 3, b = −36, c = k:

(−36)² − 4(3)(k) > 0

1296 − 12k > 0

12k < 1296

k < 108

So k can be anything below 108, and the least possible value of m is 108.

That is correct and not hard, if the discriminant is automatic. In my logs the misses on this question are almost never conceptual. They are 36² computed as 1196, or 1296 divided by 12 as 98, or the inequality flipped when dividing. The idea is fine; the arithmetic, done fast and in the head, is where the point goes. Time, done carefully: a little over a minute.

With Desmos, done wrong

A student who uses Desmos a lot types the equation as written:

3x^2-36x+k=0

Desmos offers a slider for k. The student accepts it and starts dragging, watching the parabola move, looking for the moment it has two x-intercepts. It has two for every k below some value, and the slider’s default range is −10 to 10, so nothing happens at all. Thirty seconds gone, and the student either changes the slider’s range and keeps dragging, or gives up and goes back to the discriminant.

The slider is the wrong tool for this question. It is the right tool for two-graph conditions, like a line tangent to a parabola. For a single equation with a constant, there is something better.

With Desmos, done right

Replace the constant with y:

3x^2-36x+y=0

Desmos draws a parabola opening downward. Click its top: (6, 108).

Now read the graph as a statement about k. The vertical axis is the constant. A horizontal line at height k crosses the curve where the original equation has solutions. Below the peak, the line crosses twice: two real solutions. At the peak, it touches once: one solution. Above it, nothing: no solutions. So two solutions happen exactly when k < 108, and m = 108.

Time: about twenty seconds, and most of that is the click.

Which is faster

Desmos, by about forty seconds, but that is not the main reason I teach it. The constant-as-y method has no arithmetic in it. There is nothing to square and nothing to divide, so there is nothing to get wrong in the head. And it answers every cousin of the question at once: exactly one solution is the peak, no solutions is above the peak, for which values is read straight off the axis. A student who learns the discriminant learns a formula. A student who learns this learns what the formula is saying.

Where it does not apply

The move needs an equation in one variable with one constant. If the question gives two equations, a line y = 4x and a parabola y = x² + k, and asks for the k that makes them meet exactly once, then there are two graphs and the slider is the right tool: type both, accept the slider, drag until the curve just touches the line. Or combine them first into x² − 4x + k = 0 and use constant-as-y again. Either works; the point is to notice which situation you are in before you type.

And it needs the student to have seen it before. Typed cold on test day it feels strange, because the y on the screen is not the y in the problem. It takes one session and a handful of drills to make it feel normal, and after that the whole question family moves from right but slow to right and fast. If your practice-test log shows the discriminant questions in the slow column, this is the fix.