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

x413x2+36=0

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

x413x2+36=0

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

(2x2)22(2x2)13+144=0

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,

20(2x2)221(2x2)13+2236=0(2x2)22(2x2)13+144=0

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:

(2x2)22(2x2)13+144+25=0+25(2x2)22(2x2)13+169=25

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: (ab)2=a22ab+b2. There is also a a+b version.

(2x2)22(2x2)13+169=25(2x213)2=25

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:

(2x213)2=252x213=±5

Step F - Re-arrangePermalink

Let’s isolate the x2 like it has covid:

2x213=±5x2=13±52

Step G - Almost there!Permalink

Now we take the square root remembering our ±:

x2=13±52x=±13±52

Step H - The answers!Permalink

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

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