Several times over the past few weeks we have talked about the fact that we could only find the zeros of a quadratic if we could factor it. Today we found ways of dealing with the case when it does not factor.
As a class they were struggling with this so I opened up Desmos and asked for some values. One student gave me an equation with decimal values for 'b' and 'c' which gave zeros that would not easily have been found algebraically. This, once again, set up the need to find another way to solve quadratics.
We worked through a couple of examples where the equation was in vertex form. I told them that one of the big challenges was knowing when to expand. In these cases, solving would be easier if we did not expand.
This proved to be a great opportunity to review the effect of the 'a' value when graphing. We went back to our pattern of "from the vertex, go right/left 1, up 1; right/left 2, up 4; right/left 3, up 9" to graph each of these parabolas. The algebraic solution, although new, seemed to make sense.
Okay - so now given an equation in standard form, they could complete the square to get it in vertex form, then solve as we did above. I told them we could generalize the process and then I did. The curriculum says that students should be able to follow the development of the quadratic formula, not replicate it, so it was all pencils/pens down as we worked through a case with numbers alongside the general equation.
As is often the case, students were generally unimpressed with this "ugly" formula. I asked if they would rather complete the square then solve each time they could not factor or simply substitute values into the formula. Some were sold.
Then we practiced with three particular questions.
It took a little questioning to get them all to see that what was under the square root was the determining factor in the number of solutions. I had a Desmos file ready to go, but felt that they understood how the discriminant showed the number of roots so I skipped it. We did a couple of examples to be sure.
Then we started on a more interesting question. The actual calculations are not difficult but choosing what tool to use is.
Here is yesterday's homework and here is today's.