Use questions about areas to motivate the quadratic equation, like:
- What are the dimensions of a rectangle with a known area and perimeter
- What are the dimensions of a rectangle with a known area and difference between side lengths
- What are the dimensions of a rectangle with a known area and diagonal length
(and perhaps others? any ideas?)
Then form the general quadratic equation and try to solve it.
I'm not really sure how one would discover the solution so maybe someone has an idea?
Then somehow we see that if we allow for negative numbers then these are all of one "kind" but if we don't there are three different kinds of quadratic equations with different solutions. So we would like to allow for negative numbers that preserve the same properties as positive numbers -- thus negative times negative is positive.