Some students WANT to know WHY. I use methods similar to the example below to meet that requirement........
Find where the following functions intersect: y=2x^2 and y=7x-5
The values of the two functions above are equal where they intersect; to find this point of intersection, we set the functions equal to one another. The result is 2x^2=7x-5.
We now set this equation equal to zero: 2x^2-7x+5=0.
This next part takes care in explaining. Since we want to solve for x^1 (which is linear), our goal is to now express the left side of this equation as the PRODUCT of 2 linear functions. We want the product of these two linear functions to equal 0; this only happens if one OR the other is equal to “0″. THIS is why we must first set the equation equal to "0" and then factor the quadratic; break it into the product of linear functions. Since this product must equal zero, one of those linear functions must equal zero to satisfy the given condition given by (2x-5)(x-1)=0. The solution to this equation is therefore x=5/2 or x=1.
Once this first example has been completed, I like making a small change in the original functions that ties in with my notes on transformations. These can be viewed by clicking on the link below.
Transformations (Quadratic Functions)
Here is what I like to do next; replace “x” in each of the two original functions with “(x+3)". This, as you should know, will move everything to the left by 3 units (including the solutions). Those solutions should therefore be x=-1/2 and x=-2.
Now we can verify these solutions. This is a valuable exercise to work through with students as it allows them to visualize the geometric representation AND use REASON to find solutions (or at the very least, to predict what should happen). We need to teach transformations anyways; why not incorporate some of that here while we’re at it.
Let’s solve the new system:
y=2(x+3)^2 and y=7(x+3)-5
2x^2+12x+18=7x+16 ==> 2x^2+5x+2=0 ==> (2x+1)(x+2)=0 ==>x=-1/2 & x=-2
Represent everything graphically and connect the algebraic representation of each to the corresponding geometric version; I believe this empowers students AND motivates them to learn more.
I don’t spend weeks on this type of thing; in fact, this goes very quickly. Once these basic notions are understood by students, I DO use graphing technology; they do NOT have to be mutually exclusive. I’m certain we all want the same for our students and there will always be more than one effective approach. I think the approach I’ve shown here and in my notes on facebook have value in helping kids have a better UNDERSTANDING of math.
The logic behind this process is identical for higher degree polynomial equations; in fact, the same thought process can easily be transfered to exponential, logarthmic, trigonometric equations and others. This becomes very powerful when adopted and my students surprise themselves at how easy these so-called "complicated" tasks become. Representing higher order polynomials as the product of linear "functions" will be carried out at much higher levels of mathematics as well. An example of this can be seen by clicking on the link directly below; the notes contained within relate to the process of Integration by Partial Fractions, which is part of my Introduction to Calculus.
Integration by Partial Fractions
James Tanton has made available his view on how QUADRATICS should be taught; a link to this is found below. It would benefit everyone studying quadratic functions to have a good look at the pages contained within.
James Tanton - Quadratics