While working on quadratic equations with students I have discovered a few techniques that are particularly effective. By far the most effective is to require the students to solve each one by all three methods ( factoring, completing the square and quadratic
formula ) for each and every problem rather than solving it only by the easiest way and to require the graph for each and every one. Of course, most quadratics are more easily solved by one particular method rather than the other two so I allow them to do
the easiest first and simply prove the result with the other two. This technique assures that the student can do it in each way and that they develop the skill of determining which is the “best” way for any particular problem. Another is to require students
to show each and every quadratic in both standard form ( ax^2+bx+c ) and in vertex form (a(x-h)+k). Still another is to explain and require students to be able to explain the derivation of the quadratic formula. These methods help students to understand the
concepts globally and thence to master the subject. Once students have mastered quadratics I ask them to solve them “in their head”, that is, without the use of paper or pencil ( Once they can tell me the vertex, x intercepts, y-intercept and general shape
I ask that they graph it.) NOTE: most students do not believe, initially, that they can do these without paper & pencil and are happily astounded at their own ability when they succeed . . . a very happy result. Once they ARE able to do this I have them do
many in this way and then prove their answers with paper & pencil ( and that’s how I recommend to them to approach test problems thenceforth ).