Chapter 1: Introduction to the Equation

Imagine sitting in a high school math exam where you need to validate the equation:

You might easily verify this by expanding the brackets on the left side. For instance:

And that's the proof concluded.

But what if I suggest that we can demonstrate that this equation holds true solely for the values x = 1, x = 2, and x = 3? For example:

Is it incorrect to assert that this is enough to prove the equation holds? Allow me to explain!

Chapter 2: A Unique Proof

Let’s take a step back and momentarily set aside the idea that this equation is valid for every possible value of x.

What if we consider that it is only valid for certain x values? This perspective allows us to treat either side of the equation as distinct quadratic functions of x.

Visually, the equation searches for the common points, or "intersections," between two parabolas. This allows us to rewrite the equation (by transferring the right side over to the left and adjusting for constants a, b, and c):

As Equation 2 is quadratic, it can have a maximum of two solutions. This is akin to stating that two different parabolas can intersect at most twice (e.g., see the two intersections in the diagram below—feel free to challenge this by illustrating a case with more than two intersections!).

However, we have already established that there are at least three x values (0, 1, and 2) that satisfy Equation 2! This indicates there are at least three intersection points between the two parabolas, which can only occur if the two supposedly distinct parabolas are, in fact, identical.

End of proof.

In broader terms, this proof can be applied to higher order equalities. For instance, to validate equalities of order n, you only need n + 1 specific examples.