Square root of simple binomial function

Question

Let's say I have the following formula:

$$\sqrt{a^2-2ab+b^2}=\sqrt{(a-b)^2}=\sqrt{(b-a)^2}$$

When do I know which one of the following I should use?:

$$\sqrt{(a-b)^2}=a-b\qquad\text{ or }\qquad \sqrt{(b-a)^2}=b-a$$

Answer 1

$\sqrt{(a-b)^2} = |a-b| = |b-a| = \sqrt{(b-a)^2}$.

Without the absolute value sign, the identity is correct only when the larger of the two numbers comes first in the subtraction, since the radical refers to the nonnegative square root.

Answer 2

Let $x=a-b$, then $-x=b-a$. Then $$\sqrt{(b-a)^2}=\sqrt{(-x)^2}=\sqrt{x^2}=\sqrt{(a-b)^2}$$ Now matter what real number you take for $a$ and $b$, you always have $$\sqrt{(b-a)^2}=\sqrt{(a-b)^2}$$

So in my opinion, your question may be somewhat misleading for yourself. You are actually asking when $$\sqrt{x^2}=x$$ and when $$\sqrt{x^2}=-x.$$ So what you need is the definition of square root: for all real numbers $x$, $$\sqrt{x^2}=|x|$$ Now you can go on the argument yourself.