square root solutions

Question

Is there a specific rule to get square root of any non-negative number?. The main reason why I'm asking this is that my maths teacher told me there is only one solution can be contained for any non-negative number and it always be the positive solution. Do you agree with that?

Answer 1

Generally speaking, if $a > 0$ then:

There is only one value for $\sqrt{a}$.

There are two solutions for the equation $x^2 = a$, namely: $\sqrt{a}$ and $-\sqrt{a}$.

Answer 2

The function $\sqrt x$ is designed to give you a number which square equals $x$. However you can always find two such numbers. To make $\sqrt x$ into a function you want its value to be unambiguously defined. So we use a convention:

(*) "For a positive real number $x$, $\sqrt x$ is the positive number $b$ such that $b^2=x$.

So yes I do agree with your teacher. That is because your teacher agrees with the rest of the mathematical society, that though for any non-negative real $x$ there are two possible values for $y$ such that $y^2=x$; we define $\sqrt x$ by the convention (*).

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For negative values of $x$, the value of $\sqrt x$ will be purely imaginary. So here the words positive and negative, formally don't have a meaning. However each purely imaginary number can be uniquely written as: $bi$, where $b$ is a real number. In this case we use this convetion:

"For a negative real number $x$, $\sqrt x$ is the purely imaginary number $bi$ such that $(bi)^2=x$ and $b\geq0$.