Question

How do I doD%20" id="TexFormula1" title=" \sqrt{9 - n = \sqrt{ \frac{n}{2} } } " alt=" \sqrt{9 - n = \sqrt{ \frac{n}{2} } } " align="absmiddle" class="latex-formula">
I need to find the value of n​

Answer 1

I'm assuming that there's a typo and you mean

$\sqrt{9-n} = \sqrt{\dfrac{n}{2}}$

We must use two key facts: the square root is defined only for non-negative inputs, so we must require

$\begin{cases}9-n\geq 0\\\frac{n}{2}\geq 0\end{cases} \iff \begin{cases}n\leq 9\\n\geq 0\end{cases}$

So, we will only accept values in the range

$0 \leq n \leq 9$

Secondly, where defined, the square root is an injective function. This means that you can only have two equal outputs if you start from two equal inputs:

$\sqrt{x}=\sqrt{y}\iff x=y$

So, in your case, we have

$\sqrt{9-n} = \sqrt{\dfrac{n}{2}} \iff 9-n = \dfrac{n}{2}$

Adding n to both sides gives

$9 = \dfrac{3n}{2} \iff 3n = 18 \iff n = 6$