I know from the pictures below how to interpret the binomial theorem and it makes sense. But I can't make sense out of it if the equation I'm expanding with it is a function. For example if I have a function:
$f(x)=x^2 -2x+1$
Now if I want to find it's roots I equate it to zero and solve for x.
$0=x^2-2x+1$
I can do that using the binomial theorem for two dimensions. $(a+b)^2=a^2+2ab+b^2$
The geometric interpretation is below. Now using this the function becomes
$0=x^2-2x+1=(x-1)^2$
And I know that it becomes zero if x=1 that's fine. But what is the geometric interpretation of that in terms of the pictures below? You have a function, and not everything is constant, so you don't calculate the area of a square. What is the geometric interpretation that you can find the roots with it??
The function you describe is $f(a)=a^{2}-2ab+b^{2}$ when $a=x$ and $b=1$. Look at the picture below:
It shows that $(a-b)^{2}$ is the area of the light brown square. And it is equal to $a^{2}-b^{2}-2(a-b)b=a^{2}-b^{2}-2ab+2b^{2}=a^{2}-2ab+b^{2}$.
The closer $b$ to $a$, the more little $(a-b)^{2}$, the area of the square of side $(a-b)$. It is then natural that $(a-b)^{2}$ is equal to $0$ when $a=b$, because it is the area of a square of side $0$.