I got the question:
How many solutions for $x = \sqrt 9$?
the options are: A. 1, and B. 2
However, I'm confused whether the solution is only $+ 3$ or $\pm3.$
Someone has said that $x = \sqrt9$ is not equal with $x^2=9,$ therefore the answer is only 1, it's positive 3.
Why $x = \sqrt9$? is not equal to $x^2 = 9$ ? Can't I change it with this method:$\implies(x=\sqrt 9)^2\\ \implies x^2= 9$
Am I missing something here?
The problem here is mainly notational. There is only one solution to $x=\sqrt9$, which comes from the way that the square root function is defined. We often learn in grade school that $\sqrt x$ is defined as "the number whose square is equal to $x$", but this definition is incomplete: the square root function is defined as the ''positive'' number whose square is equal to $x$. As a result, even though $3^2=9$ and $(-3)^2=9$, it is incorrect to say that $\sqrt9=-3$. Mathematicians could have defined the square root function to map a number to both its positive and negative square roots (in other words, a function that maps a number to an ordered pair of numbers) but they did not.
As a side note, your manipulation of the equation $x^2=9$ to $x=\sqrt9$ is an example of what is known as an "extraneous solution." These kinds of solutions can crop up whenever you try to prove a statement in reverse, by starting with what you want to prove and manipulating it until you get a statement of fact. Whenever you prove something this way, there are some functions (including $\sqrt$) which may get you an answer with no real meaning. In other words, be careful whenever you try to do backwards proofs!
Hope this helps.