Question

what is the solution to the equation \\(\sqrt{2n + 28} - 4\sqrt{n} = 0\\)?
\\(n = 4\\)
\\(n = 14\\)
\\(n = 2\\)
\\(n = 7\\)

Explanation:

Step1: Isolate the square root term

Start with the equation \(\sqrt{2n + 28}-4\sqrt{n}=0\). Add \(4\sqrt{n}\) to both sides to get \(\sqrt{2n + 28}=4\sqrt{n}\).

Step2: Square both sides

Square both sides of the equation to eliminate the square roots. \((\sqrt{2n + 28})^2=(4\sqrt{n})^2\), which simplifies to \(2n + 28 = 16n\).

Step3: Solve for \(n\)

Subtract \(2n\) from both sides: \(28 = 16n - 2n\), so \(14n = 28\). Divide both sides by 14: \(n=\frac{28}{14}=2\)? Wait, no, wait, let's check again. Wait, maybe I made a mistake. Wait, let's substitute the options. Let's try \(n = 14\): \(\sqrt{2*14 + 28}-4\sqrt{14}=\sqrt{28 + 28}-4\sqrt{14}=\sqrt{56}-4\sqrt{14}=2\sqrt{14}-4\sqrt{14}=-2\sqrt{14}
eq0\). Try \(n = 4\): \(\sqrt{8 + 28}-4\sqrt{4}=\sqrt{36}-8 = 6 - 8=-2
eq0\). Try \(n = 7\): \(\sqrt{14 + 28}-4\sqrt{7}=\sqrt{42}-4\sqrt{7}\approx6.48 - 10.58
eq0\). Wait, wait, my step 2: \((4\sqrt{n})^2 = 16n\), and \(\sqrt{2n + 28}\) squared is \(2n + 28\). So \(2n + 28 = 16n\), then \(28 = 14n\), so \(n = 2\)? But the options have \(n = 14\), \(n = 4\), \(n = 2\), \(n = 7\). Wait, maybe I misread the original equation. Wait, the original equation is \(\sqrt{2n + 28}-4\sqrt{n}=0\)? Wait, maybe it's \(\sqrt{2n}+28 - 4\sqrt{n}=0\)? No, the user wrote \(\sqrt{2n + 28}-4\sqrt{n}=0\). Wait, let's check \(n = 14\) again: \(\sqrt{2*14 + 28}=\sqrt{28 + 28}=\sqrt{56}=2\sqrt{14}\), \(4\sqrt{14}\), so \(2\sqrt{14}-4\sqrt{14}=-2\sqrt{14}
eq0\). Wait, maybe the equation is \(\sqrt{2n}+28 - 4\sqrt{n}=0\)? No, the user's equation is \(\sqrt{2n + 28}-4\sqrt{n}=0\). Wait, maybe I made a mistake in solving. Let's do it again. Step 1: \(\sqrt{2n + 28}=4\sqrt{n}\). Square both sides: \(2n + 28 = 16n\). Then \(28 = 14n\), so \(n = 2\). But the option has \(n = 2\). Wait, the options are \(n = 4\), \(n = 14\), \(n = 2\), \(n = 7\). So \(n = 2\) is an option. Wait, but when I substitute \(n = 2\): \(\sqrt{4 + 28}-4\sqrt{2}=\sqrt{32}-4\sqrt{2}=4\sqrt{2}-4\sqrt{2}=0\). Ah! There we go. So \(n = 2\) is the solution. Wait, the options have \(n = 2\) (the third option). So the correct answer is \(n = 2\).

Answer:

\(n = 2\) (the option with \(n = 2\))