This week I was making up systems of quadratic equations for my kids to solve by graphing. In my experience this is the sort of task that seems like it should be easy but which I have spent a number of years doing poorly, carelessly, and at length!
Here's my new trick:
1. Choose two binomial factors whose products will have integer roots, i.e. (x + 3)(2x - 4)
2. Multiply those factors.
3. Set the product equal to zero.
4. Use inverse operations, your own creativity and the principle of equality to move some part of each term to the other side.
5. Those are your equations, and the solutions will be {-3, 2}, the roots of your original quadratic.
Voila!
At this point you get to play with how nice the roots and vertices of your quadratics are, but it doesn't matter much to me for solving systems by graphing. Of course then you can plug your equations into Wolfram Alpha to check yourself.
Hope this helps!
Great trick. I will pass this on to our workshop teachers this summer!
ReplyDeleteThis is fantastically genius. Thank you. Now I need a way to make up equations with rational expressions in them on the fly that have nice solutions, please. :)
ReplyDelete@Kate I generally do the same thing twice: one polynomial picks out zeroes, one picks out poles, and
ReplyDeleteI can tweak it to adjust asymptotes, or to complicate matters by making factors that cancel out.