0

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?

Arvin
  • 3
  • 2
    There is a difference between asking to solve an equation like $x^2=9$ (which has two solutions) and plugging $x=9$ into the function $f(x) =\sqrt{x}$ (which has only one output). Also, $\sqrt{x^2} = |x|$ (not $x$); while $(\sqrt{x})^2=x$ only makes sense for $x \geq 0$. – Randall Dec 08 '22 at 15:56
  • 4
    Squaring both sides of an equation does not give an equivalent equation. For example 1=-1 is false, but if you square both sides, it is true. This is why you have to check solutions to the new equation to see if they solve the original. – Cheerful Parsnip Dec 08 '22 at 15:59
  • See this related question. – N. F. Taussig Dec 08 '22 at 16:00
  • Welcome to MathSE. This Mathjax tutorial explains how to typeset mathematics on this site. – N. F. Taussig Dec 08 '22 at 16:06
  • 1
    It partly depends on notation, but since $\sqrt 9$ is a functional notation, we usually adopt a reading that $\sqrt 9$ takes a single value. – Thomas Andrews Dec 08 '22 at 16:09
  • 1
    As for squaring: If you started with $x=1,$ and square both sides, you get $x^2=1,$ so is $x=-1$ also a possibility? No, squaring adds a new solution, $-1,$ that wasn't present in the original solution $x=1.$ – Thomas Andrews Dec 08 '22 at 16:10
  • 1
    Also, where you are at the last line, I think you meant $x^2=9,$ not $x^2=\sqrt9.$ – Thomas Andrews Dec 08 '22 at 16:13
  • 2
    A few asides: we don't write that two statements are "equal," we write that they are "equivalent." And we try to avoid abuse of notation like $(a=b)^2,$ since $a=b$ represents a statement of equality, not a number, and a statement cannot be squared. – Thomas Andrews Dec 08 '22 at 16:16
  • It is a convention that $\sqrt{x}$ is always the non-negative possibility. If we mean both solutions of $w^2=x$ , we usually write it as $w=\pm \sqrt{x}$ which is a slight abuse of notation. But this way , for example , the formula to solve quadratics can better be memorized. – Peter Dec 08 '22 at 16:19
  • @N.F.Taussig Your Mathjax tutorial reference helps me a lot in typeset mathematics, thank you.

    Also I've got some understanding for the square root functions now, thanks for all the clear explanations guys.

     – Arvin Dec 08 '22 at 17:10

1 Answers1

2

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.

slithy_tove
  • 72
  • The positive number , unless $x=0$ , in which case there is only one possibility and we simply have $\sqrt{0}=0$ – Peter Dec 08 '22 at 16:24
  • A map cannot map some number to more than one number. To establish a map , one of the branches must be chosen, it was a natural choice to choose the "positive branch". – Peter Dec 08 '22 at 16:25
  • 1
    Thank you for your explanation. Now I just learned that the square root formula is $\sqrt x^2 = |x|$ Hence why the solution is only one, the positive ones. – Arvin Dec 08 '22 at 17:08