A student assumed that a non-factorable quadratic had no solution.

Several were surprised when quadratic formula and completing the square yielded the same solution.

I came up with this little analogy to help clear things up, and it ended up working pretty well.

I started out by talking about how students might drive to Sonic, a favorite fast food place that is 10 miles away from our tiny town. Students named several paths and we talked about their pros and cons. The interstate is fast, but it does not provide a direct path to Sonic. Most students would choose to take Old 40 highway, but it is well known for road construction and can be blocked for days/weeks/months at a time. Someone even pointed out that there is a dirt path through a corn field that many of us have used to bypass the road construction. Perfect.

Solving a quadratic equation is very much the same.

I started out by talking about how students might drive to Sonic, a favorite fast food place that is 10 miles away from our tiny town. Students named several paths and we talked about their pros and cons. The interstate is fast, but it does not provide a direct path to Sonic. Most students would choose to take Old 40 highway, but it is well known for road construction and can be blocked for days/weeks/months at a time. Someone even pointed out that there is a dirt path through a corn field that many of us have used to bypass the road construction. Perfect.

Solving a quadratic equation is very much the same.

On the left you have your quadratic equation. You want to get over to the solution, on the right. Factoring, complete the square, and quadratic formula give you three potential paths to get there.

Sometimes, when you take the factoring route, you find that your equation can't be factored. When this happens, its like a road closed sign on the path to your solution. The solution is still there, but you'll have to take a different route to get to it.

Other times, when you're completing the square, you run into a bunch of fractions. We aren't afraid of fractions here, but they can turn an otherwise straight path to the solution into a long and winding road. If we're looking for the most efficient way to get there, we want to avoid this situation.

Then there's the quadratic formula. It may not always be the most efficient, but it always works. It is particularly useful for avoiding the road block and the winding road.

My favorite quadratic formula sign said "when you don't want to think about which road to take". They're not wrong.

For a follow up activity, this card sort would be perfect.