Simplifying radicals is strangely satisfying.

I’m helping my son with mathematics, and I have to re-learn it, but this is fun. Hence, I’m hoping to start a series of articles.

Problem statement #

Given:

Show that:

Solution #

Starting with:

Often such sums can be rewritten like this:

We observe the formula:

Therefore, we end up with the following system of equations:

The solutions may or may not jump at you, but there’s a little Vieta’s formula that can help — we can find a and b as being the solutions to this quadratic equation:

Solving that:

Threfore:

Going back to our original value, we rewrite it:

We still have a square of a sum, but this time we can spot the formula easily:

Next, for simplifying A, we’ll focus on rewriting this:

Same trick applies, so we’re dealing with this system:

Which will have solutions generated by this quadratic equation:

Solving it:

Therefore our sum can be rewritten as (we remember that we have a minus sign though 🙂):

We can now rewrite our A:

And finally: