Let's solve the equation step-by-step: [tex]\(\sqrt{2n + 28} - 4\sqrt{n} = 0\)[/tex].
1. We start with the given equation:
[tex]\[
\sqrt{2n + 28} - 4\sqrt{n} = 0
\][/tex]
2. To isolate [tex]\(\sqrt{2n + 28}\)[/tex], let’s make [tex]\(\sqrt{2n + 28}\)[/tex] equal to [tex]\(4\sqrt{n}\)[/tex]:
[tex]\[
\sqrt{2n + 28} = 4\sqrt{n}
\][/tex]
3. Now, we square both sides of the equation to remove the square roots:
[tex]\[
(\sqrt{2n + 28})^2 = (4\sqrt{n})^2
\][/tex]
Simplifying this, we get:
[tex]\[
2n + 28 = 16n
\][/tex]
4. We need to solve for [tex]\(n\)[/tex]. Let’s isolate [tex]\(n\)[/tex] on one side of the equation:
[tex]\[
2n + 28 = 16n
\][/tex]
Subtract [tex]\(2n\)[/tex] from both sides:
[tex]\[
28 = 14n
\][/tex]
5. Finally, divide both sides by 14:
[tex]\[
n = \frac{28}{14}
\][/tex]
[tex]\[
n = 2
\][/tex]
So, the solution to the equation [tex]\(\sqrt{2n + 28} - 4\sqrt{n} = 0\)[/tex] is [tex]\( n = 2 \)[/tex]. Therefore, from the given options, [tex]\( n = 2 \)[/tex] is the correct answer.
