Square-Root of a binomial squared

[lightlightsup]

Homework Statement

$\sqrt{(a-b)^2}=\sqrt{4x^2}$
$±(a-b)=±2x$
Is this a correct distribution of the $±$s?

Relevant Equations

None.

Suppose that $a$, $b$, and $x$ are integers.
How would the $±$s be correctly assigned in such an equation?

 

[PeroK]

The solutions to that equation are simply $2x = \pm (a-b)$.

 

[HallsofIvy]

It really doesn't matter. If we were to accept $$\pm a= \pm b$$ literally there would be four cases:

1) taking "+" on both sides, a= b
2) taking "-" on both sides, -a= -b
But this is just a= b with both sides multiplied by -1. They are the same.
3) taking "+" on the left side and "-" on the right, a= -b
4) taking "-" on the left side and "+" on the right, -a= b
But, again, the second is the same as the first, just with both sides multiplied by -1.

So there are really just two different cases which can be given by either $a= \pm b$ or $\pm a= b$.

 

[SammyS]

Homework Statement:: $\sqrt{(a-b)^2}=\sqrt{4x^2}$
$±(a-b)=±2x$
Is this a correct distribution of the $±$s?
Relevant Equations:: None.

Suppose that $a$, $b$, and $x$ are integers.
How would the $±$s be correctly assigned in such an equation?

Most introductory algebra textbooks (at least those of which I am aware) point out that actually, $\sqrt{u^2 ~ }=|u|$ .

So for instance, solving an equation such as, $t^2=9$, by taking the square root of both sides gives:
$\sqrt{t^2\ }=\sqrt{9\ }$

simplifying gives: $|t|=3$ .
So that if $t > 0$, then $|t|=t$ thus we have $t = 3$ .
However, if $t < 0$, then $|t|=-t$ thus we have $-t = 3$ .

This gives the solutions: $t = 3$ or $t=-3$ .

I tend to think of the $\pm$ method as a short cut of sorts.

 

[lightlightsup]

The solutions to that equation are simply $2x = \pm (a-b)$.

I deduced the same thing by just guessing that since both sides being positive or negative would lead to the positives/negatives cancelling out, then, the only remaining scenarios are:
one where only 1 negative sign, and
one where both sides are positive.

I was hoping for a more formal understanding of what was going on.

You guys have provided that to me.
Thank You!