This is from George Pólya’s “How to Solve It” book. I am reading this because I want to sharpen my problem solving skills and, eventually, be able to make derivations like this in Concrete Mathematics.

In How to Solve It they go through a series of equivalent problems to find the eventual solution. So we may not know the solution to our original problem A but we can go through B, C, D, etc. until we find a solution we know or that can be directly “seen”.

The book goes through solving this equation with the unknown $x$

Since I am bootstrapping my mathematical knowledge I found the book’s treatment a bit terse without many explanations of how they got to each step. I eventually figured out that they are trying to massage the equation into a state where you can apply the perfect square rule. This makes things much simpler to reason about and thus find the solution.

Step A - The original equation.Permalink

As mentioned above this is

And our task is to find $x$. Or, more specifically, where the graph intercepts the $x$ axis at $0$. The steps are essentially a set of quadratic equation equivalence conversions.

Step B - Multiple by powers of twoPermalink

My slow self needed to ponder this one for a while. In any case, B ends up being

This had me baffled for a while but I, eventually, saw that they are multiplying by increasing powers of two. I don’t know if this technique has a name or how they arrived at doing this first step but, to write it another way,

Step C - Add 25 to each sidePermalink

For those who know quadratics quite well can probably see where they are going with these steps but, anyway, I was still baffled at this stage but at least it could see the manipulation easily enough:

Step D - Make it short!Permalink

This makes things much more compact through, as I eventually figured out, the perfect square rule! To remind the forgetful like me it is: $(a-b{)}^2=a^2-2ab+b^2$. There is also a $a+b$ version.

The more compact form gives us a better handle on things. This is a key step to solve this more easily. The following steps are using this step to find the four solutions.

Step E - Partial “solution”Permalink

Here we take the square root and find a sort-of solution that still needs some work. Let’s call it the IKEA solution:

Step F - Re-arrangePermalink

Let’s isolate the $x^2$ like it has covid:

Step G - Almost there!Permalink

Now we take the square root remembering our $\pm$:

Step H - The answers!Permalink

Now can see what the concrete answers are: $x=3$ or $-3$ or $2$ or $-2$.