How to Solve It - Chains of Auxilliary Problems
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
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
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:
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
Step G - Almost there!Permalink
Now we take the square root remembering our
Step H - The answers!Permalink
Now can see what the concrete answers are: