Can x be written as both a positive and negative value in an equation?

[LiHJ]

<< Mentor Note -- OP has been advised to type their questions into the forum next time, instead of inserting images >>

1. Homework Statement
Dear Mentors and PF helpers,

I have a question from today's lesson.

Homework Equations

The Attempt at a Solution

Is 1) and 2) both acceptable? If only of of them is acceptable, is there any reason why?

Thanks for your time

 

[RUber]

Only 2.
The $\sqrt{}$ implies that it only returns the positive square root of a number. That is why you will often see $\pm \sqrt{}$ in formulas.

 

[HallsofIvy]

For x any real number, $\sqrt{x^2}= |x|$, NOT $\sqrt{x}= x$

That is easily seen by taking x to be any negative number, say x= -2. In that case, $x^2= (-2)^2= 4$ and $\sqrt{x^2}= \sqrt{4}= 2$, NOT -2.

 

[RUber]

However, if you know something about x, like it is positive, then you may use the notation in (1) .
Also, if you have $y = x^2$, then $x = \pm \sqrt{y}$.