Sure, it's short. But it doesn't get to why one would do these steps if you don't know they are going to lead to a solution. Completing the square is a non-obvious step to make.
The argument in my previous comment attempts to provide motivation for such a step, starting from simpler questions. It uses the geometry of the problem, and builds up from solving a simpler problem first.
That approach also uses the notion of transformation and invariance (seeing what happens to the roots when we move the graph around, and noticing that the distance between the roots doesn't change we shift horizontally).
Again, the important part here, is that you could lead someone to ask the same questions and have them answer them themselves.
"Let's solve this equation. Looks complicated. Can we solve a simpler problem first? What would a simpler problem be? Now how can we make it a little more complicated, and how does it affect the answer?".
And that's how math is done.
After all is said and done, one can extract "completing the square" as a shortcut technique. But that's what it is - a shortcut through the woods. Learning a shortcut won't teach you how to walk in the forest on your own.
The argument in my previous comment attempts to provide motivation for such a step, starting from simpler questions. It uses the geometry of the problem, and builds up from solving a simpler problem first.
That approach also uses the notion of transformation and invariance (seeing what happens to the roots when we move the graph around, and noticing that the distance between the roots doesn't change we shift horizontally).
Again, the important part here, is that you could lead someone to ask the same questions and have them answer them themselves.
"Let's solve this equation. Looks complicated. Can we solve a simpler problem first? What would a simpler problem be? Now how can we make it a little more complicated, and how does it affect the answer?".
And that's how math is done.
After all is said and done, one can extract "completing the square" as a shortcut technique. But that's what it is - a shortcut through the woods. Learning a shortcut won't teach you how to walk in the forest on your own.