## (x+y)² or (x+y)^2

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© Copyright 1999, Jim Loy

Some people might guess that (x+y)²=x²+y². But, if you are reading this article, I suspect that you know that:

(x+y)² = x²+2xy+y²

How do we know that? Let's see:

(x+y)² = (x+y)(x+y)
= x(x+y) + y(x+y)    {distributive law)
= x²+xy  + xy+y²
= x²+2xy+y²

Not bad. You might want to verify, for yourself that the second step is an application of the distributive law, which says a(x+y)=ax+ay.

There is a way to make this more obvious. To the left is a square. Each side is x+y. What is the area? Well, it is (x+y)². But, we can also find the area, by summing up the areas of the four rectangles (two of which are squares) which make up our big square:

area = x²+xy+xy+y²
= x²+2xy+y²

Beginners should find that easy to understand.

Here is a magic trick. Use a calculator with square roots, for this:

• Pick a number, any number. You may have to remember it, below.
• Square it.
• Add twice the original number.
• Add one.
• Take the square root of this.
• Subtract your original number.
• I bet your answer is 1 (or maybe .999999...)

The answer may be .999999..., if your number was not an integer, because of round off error.

The trick involved building up x²+2x+1, which we know is (x+1)². So, then we can take the square root, and subtract x, to get 1.

When I was in school, I occasionally (only four or five times) was marked off for writing things like .25, when I should have written 0.25. 0.25 is a little clearer, as you may not notice the decimal point with .25. But, many many math books leave out the leading "0".

I once overheard an argument. One person said that .1 was 1/10. The other person said that .1 was 10%. Of course, they were both right.