Many conclusions here but would you want your students to know why 'a' MUST EQUAL 3 and 'b' MUST EQUAL -2.
COREFLECTIONS
So what's the BIG IDEA here? Is this really "Fundamental"? Where is it in the Common Core?
So if a polynomial equation of "degree" n has more than n solutions, what exactly does that imply? Any restrictions on the coefficients? And what does this have to do with an identity?
For me, it's critical that we don't see these problems as curiosities or challenges designed for only the accelerated groups or the mathletes.
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One question you can ask yourself is: How do I interpret the given (e.g., how do I record it symbolically?), and what conclusions might I draw?
For example, the given is that there are at least two values of x for which the expression on the left is 0. What does this mean?
Well, it means there are distinct N, M for which:
(a-3)N + (b+2) = 0, and
(a-3)M + (b+2) = 0.
Introducing that bit of notation is nontrivial in and of itself!
And now? Perhaps we can see what the two equations tell us.
In fact, subtracting yields:
(a-3)(N-M) = 0.
But N and M are distinct; so we need a-3 = 0, i.e., a = 3.
Working with our original given,
(a-3)x + (b+2) = 0 becomes
b+2 = 0,
whence b = -2.
Recording our thinking with non-ambiguous notation is a Big Idea, for sure.
Nice! When math educators share ideas, several Big Ideas emerge! Your thought about the importance of training our students to represent mathematical information helped to motivate the name of this blog! I also believe that asking students if a "linear" equation of the form Ax+B=0 can have more than one solution leads them to uncover another Big Idea. Having them think about this both algebraically and graphically deepens that understanding. Generalizing this idea to quadratic functions is a logical next step. Can a quadratic have THREE ZEROS? Interpret that graphically!
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