# Another stupid word problem

Jan Hlavacek · 2019/12/10 · 1 minute read

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?