To solve the equation \(\sqrt{2n + 28} - 4\sqrt{n} = 0\), we'll proceed step-by-step:
1. Isolate one of the square root terms:
First, let's isolate \(\sqrt{2n + 28}\) on one side of the equation:
[tex]\[
\sqrt{2n + 28} = 4 \sqrt{n}
\][/tex]
2. Square both sides:
To eliminate the square roots, we'll square both sides of the equation:
[tex]\[
(\sqrt{2n + 28})^2 = (4 \sqrt{n})^2
\][/tex]
Simplify the equation:
[tex]\[
2n + 28 = 16n
\][/tex]
3. Solve for \(n\):
Rearrange the equation to isolate \(n\):
[tex]\[
2n + 28 = 16n
\][/tex]
Subtract \(2n\) from both sides:
[tex]\[
28 = 14n
\][/tex]
Divide both sides by 14:
[tex]\[
n = 2
\][/tex]
4. Check the solution:
We should verify if \(n = 2\) satisfies the original equation. Substitute \(n = 2\) back into the original equation:
[tex]\[
\sqrt{2(2) + 28} - 4 \sqrt{2} = \sqrt{4 + 28} - 4 \sqrt{2} = \sqrt{32} - 4 \sqrt{2}
\][/tex]
Since \(\sqrt{32} = 4 \sqrt{2}\), we get:
[tex]\[
4 \sqrt{2} - 4 \sqrt{2} = 0
\][/tex]
The left side equals the right side, so \(n = 2\) is indeed a solution.
Thus, the solution to the equation \(\sqrt{2n + 28} - 4\sqrt{n} = 0\) is \(n = 2\).
Among the given options:
- \(n = 2\)
- \(n = 4\)
- \(n = 7\)
- \(n = 14\)
The correct answer is [tex]\(n = 2\)[/tex].
