Question

which equation results from isolating a radical term and squaring both sides of the equation for the equation \\(sqrt{c - 2} - sqrt{c} = 5\\)?
options:
\\(c - 2 = 25 + c + 10sqrt{c}\\)
\\(c - 2 = 25 - c\\)
\\(c - 2 = 25 + c - 10sqrt{c}\\)
\\(c - 2 = 25 + c\\)

Explanation:

Step1: Isolate the radical term

Given the equation \(\sqrt{c - 2}-\sqrt{c}=5\), we first isolate one of the radical terms. Let's isolate \(\sqrt{c - 2}\) by adding \(\sqrt{c}\) to both sides of the equation:
\(\sqrt{c - 2}=5+\sqrt{c}\)

Step2: Square both sides

Now, we square both sides of the equation \(\sqrt{c - 2}=5+\sqrt{c}\). Recall that \((a + b)^2=a^2 + 2ab + b^2\). Here, \(a = 5\) and \(b=\sqrt{c}\). So, squaring the left - hand side gives \((\sqrt{c - 2})^2=c - 2\), and squaring the right - hand side gives \((5+\sqrt{c})^2=5^2+2\times5\times\sqrt{c}+(\sqrt{c})^2=25 + 10\sqrt{c}+c\).
So, after squaring both sides, we get \(c - 2=25 + c+10\sqrt{c}\)

Answer:

The equation \(c - 2 = 25 + c+10\sqrt{c}\) (the second option from the right, or the option with \(c - 2=25 + c + 10\sqrt{c}\))