So I am grading intro algebra exams, and there is this problem:
The product of two consecutive positive even integers in 168. Find the integers.
Students are supposed to solve this by setting up and solving a quadratic equation \(x(x+2) = 168\). What’s stupid about this is that at this moment the only way to solve this equation that the students know is simple factoring, so they have to rewrite the equation as \(x^2 + 2x - 168 = 0\) and factor it. In order to factor this, they need to find two numbers whose difference is 2 and whose product is 168. In other words, they have to solve the original question.
Of course, should they know how to complete a square, they could instead rewrite the equation as \((x+1)^2 - 169 = 0\). The left side is now a difference of two squares, so itcan be factored as \((x + 1 + 13)(x + 1 - 13)\). Now we can start asking more interesting questions, for example which numbers can be written as a product of two consecutive even or odd integers?