Alright, let's work through solving for \( c \) given the equation \( 10b = 5(\sqrt{c} + 2) \).
1. Starting with the given equation:
[tex]\[
10b = 5(\sqrt{c} + 2)
\][/tex]
2. First, isolate \(\sqrt{c}\):
[tex]\[
10b = 5\sqrt{c} + 10
\][/tex]
Subtract 10 from both sides:
[tex]\[
10b - 10 = 5\sqrt{c}
\][/tex]
3. Next, solve for \(\sqrt{c}\):
Divide both sides by 5:
[tex]\[
\frac{10b - 10}{5} = \sqrt{c}
\][/tex]
4. Square both sides to solve for \( c \):
[tex]\[
\left(\frac{10b - 10}{5}\right)^2 = c
\][/tex]
Simplify the expression inside the square:
[tex]\[
\frac{10b - 10}{5} = 2b - 2
\][/tex]
Therefore:
[tex]\[
c = (2b - 2)^2
\][/tex]
5. Find the equivalent equation:
Notice in this simplified and new expression, the correct form that matches one of the given options would be:
[tex]\[
c = \left( \frac{10b - 10}{5} \right)^2
\][/tex]
This further simplifies to:
[tex]\[
c = \frac{(10b - 10)^2}{25}
\][/tex]
Therefore, the equivalent equation to find \( c \) is:
[tex]\[
c = \frac{(10b - 10)^2}{25}
\][/tex]
This matches the provided option:
[tex]\[
c = \frac{(10 b - 10)^2}{25}
\][/tex]
So the correct answer is:
[tex]\[
c = \frac{(10 b - 10)^2}{25}
\][/tex]
