# Freeing up the variables of an equation and assigning signs to them

## Homework Statement

sqrtY^2=sqrtX^2

Solving for Y we get:

Y= X and Y= -X

## The Attempt at a Solution

Since both sides of sqrtY^2=sqrtX^2 are equal I thought the equation would solve simply as Y= X. Turns out it has also the second part Y= -X.

My questions are since both sides of the equation sqrtY^2= sqrtX^2 are EQUAL

1. why don't we just get Y=X

or

2. why Y also doesn't take the negative sign just to keep the both sides of the orginal equation in balance?

Thanks.

Related Precalculus Mathematics Homework Help News on Phys.org
Possible solutions of the equation:

y = 2 and x = -2;

y = 2 and x = 2;

y = -2 and x = 2.

Mark44
Mentor

## Homework Statement

sqrtY^2=sqrtX^2

Solving for Y we get:

Y= X and Y= -X

## The Attempt at a Solution

Since both sides of sqrtY^2=sqrtX^2 are equal I thought the equation would solve simply as Y= X. Turns out it has also the second part Y= -X.

My questions are since both sides of the equation sqrtY^2= sqrtX^2 are EQUAL

1. why don't we just get Y=X

or

2. why Y also doesn't take the negative sign just to keep the both sides of the orginal equation in balance?

Thanks.
$\sqrt{y^2} = \sqrt{x^2} \Rightarrow y^2 = x^2$
$\Rightarrow y^2 - x^2 = 0 \Rightarrow (y - x)(y + x) = 0$
The solutions of the last equation are y = x and y = -x.

$\sqrt{y^2} = \sqrt{x^2} \Rightarrow y^2 = x^2$
$\Rightarrow y^2 - x^2 = 0 \Rightarrow (y - x)(y + x) = 0$
The solutions of the last equation are y = x and y = -x.
Amazing!!!

Mark44
Mentor
Why is that amazing?

Why is that amazing?
Well, because I didn't even think about factoring and finding the zeros. Simple and elegant.

X^2- Y^2=0 was given.

What I did was X^2=Y^2, then sqrtX^2=sqrtY^2 to "liberate" the variables Y and X. Then I got stuck with the signs the variable X took, what with X being both positive and negative.

So someone, in another forum, explained it with "Because Y = -X is also a solution. √(2²) = √((-2)²), for example" which was very helpful too. I just never thought about factoring.

Anyway, thank you, people!

Mark44
Mentor
If you started with x2 - y2 = 0, then the quickest approach is to factor the left side, and not messing around taking square roots.