Let's start by analyzing the given equation and the provided solutions.
The original equation given is:
[tex]\[ 10b = 5(\sqrt{c} + 2) \][/tex]
We want to solve for [tex]\( c \)[/tex].
Let's isolate [tex]\(\sqrt{c}\)[/tex]:
[tex]\[ 10b = 5\sqrt{c} + 10 \][/tex]
Subtract 10 from both sides:
[tex]\[ 10b - 10 = 5\sqrt{c} \][/tex]
Divide by 5 to solve for [tex]\(\sqrt{c}\)[/tex]:
[tex]\[ \sqrt{c} = 2b - 2 \][/tex]
Now, square both sides to eliminate the square root:
[tex]\[ c = (2b - 2)^2 \][/tex]
Hence, the provided correct equation for [tex]\(c\)[/tex] is:
[tex]\[ c = (2b - 2)^2 \][/tex]
Next, we have to determine which of the four given equations is equivalent to [tex]\( c = (2b - 2)^2 \)[/tex].
1. [tex]\( c = 10b - 10 - 5 \)[/tex]
[tex]\[ c = 10b - 15 \][/tex]
This equation does not match our derived equation, so it is incorrect.
2. [tex]\( c = (10b - 10 - 5)^2 \)[/tex]
[tex]\[ c = (10b - 15)^2 \][/tex]
This does not match our derived equation, so it is incorrect.
3. [tex]\( c = \frac{(10b - 2)^2}{25} \)[/tex]
Let's simplify this expression:
[tex]\[ c = \frac{(10b - 2)^2}{25} \][/tex]
Factor out the 10 inside the parentheses:
[tex]\[ c = \left( \frac{10b - 2}{5} \right)^2 \][/tex]
[tex]\[ c = \left( 2b - \frac{2}{5} \right)^2 \][/tex]
This is not equivalent to our derived equation, so it is incorrect.
4. [tex]\( c = \frac{(10b - 10)^2}{25} \)[/tex]
Let's simplify this expression:
[tex]\[ c = \frac{(10b - 10)^2}{25} \][/tex]
Factor out the 10 inside the parentheses:
[tex]\[ c = \left( \frac{10b - 10}{5} \right)^2 \][/tex]
[tex]\[ c = \left( 2b - 2 \right)^2 \][/tex]
This is exactly our derived equation.
So, the correct equivalent equation to find [tex]\( c \)[/tex] is:
[tex]\[ c = \frac{(10b - 10)^2}{25} \][/tex]
