Which equation results from isolating a radical term and squaring both sides of the equation \(\sqrt{c-2} - \sqrt{c} = 5\)?

A. \(c - 2 = 25 + c\)
B. \(c - 2 = 25 - c\)
C. \(c - 2 = 25 + c - 10\sqrt{c}\)
D. [tex]\(c - 2 = 25 + c + 10\sqrt{c}\)[/tex]



Answer :

To determine which equation results from isolating a radical term and squaring both sides for the equation \(\sqrt{c-2} - \sqrt{c} = 5\), follow these steps:

1. Start with the given equation:
[tex]\[ \sqrt{c-2} - \sqrt{c} = 5 \][/tex]

2. Isolate one of the radical terms:
[tex]\[ \sqrt{c-2} = 5 + \sqrt{c} \][/tex]

3. Square both sides of the equation to eliminate the square roots:
[tex]\[ (\sqrt{c-2})^2 = (5 + \sqrt{c})^2 \][/tex]

4. Rewrite the squared terms:
[tex]\[ c-2 = (5 + \sqrt{c})^2 \][/tex]

5. Expand the right side:
[tex]\[ c - 2 = 25 + 10 \sqrt{c} + c \][/tex]

With these steps, we obtain the resulting equation:
[tex]\[ c - 2 = 25 + 10 \sqrt{c} + c \][/tex]

The correct answer is:
[tex]\[ c - 2 = 25 + c + 10 \sqrt{c} \][/tex]