Due to time constraints, we had to skip completing the square, which leaves us no way to derive the quadratic formula. So today I put y = x² + 6x + 8 on the board and had the students factor it. We then found the roots and confirmed it by graphing it on Sketchpad. Then I changed the problem to y = x² + 6x + 7 and we found that we could not factor this one. I did graph the function and did show the students that there were x-intercepts, just that they were not rational. I told the students (this was after the quadratic formula quiz) that was what the formula was for – to find the roots of a quadratic even when the roots were irrational.
So I used the quadratic formula to solve y = x² + 6x + 8 (already shown at the beginning), y = x² + 6x + 7 (which we found that we couldn’t factor), then y = x² + 6x + 9 (double root) and then y = x² + 6x + 10 (no real solutions).
I then asked the students to tell me what it was about the quadratic formula gave away how many roots the quadratic had – I then wrote that:
- if b² – 4ac > 0, then there are two real roots
- if b² – 4ac = 0, then there is one real root
- if b² – 4ac < 0, then there are no real roots
I assigned questions 1, 2, 5, 6, and 7 for homework.
