Answer:
Because it wouldn't give us any additional insight in doing so.
There is a problem with the premise of your question, but I'll get to that in a minute.
The reason we don't put ± on both sides of our equation is because it wouldn't give us any additional information, and would make the equation more vague.
Let's take the example of x2=16 . The solution to this is x=±4 , and your question is essentially "Why don't we say ±x=±4 ?" Well, let's say that we did. To get all the possible values for x , we will look at each possible combination of positive and negative for the left and right sides of the equation. We will get the following:
x=4
x=−4
−x=−4⟹x=4
−x=4⟹x=−4
Now you technically have 4 answers, but you'll notice that (1) is actually equivalent to (3), and (2) is equivalent to (4). So now we have 4 expressions, to represent only 2 possible values of x (ie 4 and −4 ).
I also mentioned earlier that there was a problem with the premise of your question, which I will address now. A square root is actually always defined as positive. x2−−√=|x|≠±x . This is a common misconception I see a lot of people come to. The fact that both the positive and negative number solutions exist comes from the fact that we know that the sign (positive or negative) of x does not affect the sign of x2 , since x2 will always be positive (assuming x∈R ). So 16−−√≠±4 . However, (x2=16)⟹(x=±4) . But I digress...
So knowing all of that, let's look at the pros and cons of putting the ± sign on both sides of the equation, and try to speculate why that is not the standard.
l hope it's help
