Chapter 1: The Challenge of Solving

Have you ever encountered a challenging math problem? In this article, we delve into an intriguing algebraic question that has sparked curiosity among my students. This is part of an ongoing series where I explore the fascinating queries raised by learners. If you're interested in exploring all the questions from this series, along with insights into my motivation for starting it, click here.

The problem is stated as follows:

I encourage you to tackle this problem on your own before looking at the solution.

Solution

First, you might notice that one solution is a=1. The real inquiry, however, is whether additional solutions exist. While various methods could be employed to approach this question, I have encountered similar problems before, and they often hinge on a crucial observation.

Using this insight, we can reformulate our equation as shown below.

At this point, the equation still appears somewhat unwieldy, so let's simplify it with the following substitution:

This leads us to the new equation:

Next, we need to solve this quadratic equation. Applying the quadratic formula yields:

Thus, we find:

Substituting back in for x presents us with two scenarios. Let's examine each one individually.

The first case yields the solution we previously identified. The second case provides us with:

In summary, we have identified exactly two solutions for the equation: a=1 and a=-1. As mentioned, this is a relatively straightforward problem once you recognize the underlying trick. I hope this insight will aid you in tackling similar challenges in the future!

Thanks for taking the time to read this article. If you enjoy the content, I would greatly appreciate your support by following my publication, Y(Math), to help spread the word.

The first video titled "The Math Question that's Breaking the Internet! Can You Solve It?" dives into a viral math challenge that has captivated many.

The second video, "What To Do When a Student Asks a Question and You Don't Know the Answer!" provides valuable strategies for educators facing tough questions.