Let's start with the given equation:
[tex]\[ 10b = 5(\sqrt{c} + 2) \][/tex]
We will solve for [tex]\( c \)[/tex] step-by-step.
Step 1: Isolate the term involving [tex]\( c \)[/tex]
Divide both sides by 5 to simplify:
[tex]\[ \frac{10b}{5} = \sqrt{c} + 2 \][/tex]
[tex]\[ 2b = \sqrt{c} + 2 \][/tex]
Step 2: Isolate the square root of [tex]\( c \)[/tex]
Subtract 2 from both sides:
[tex]\[ 2b - 2 = \sqrt{c} \][/tex]
Step 3: Square both sides to eliminate the square root
[tex]\[ (2b - 2)^2 = c \][/tex]
We have found:
[tex]\[ c = (2b - 2)^2 \][/tex]
To find an equivalent form of this equation, we can express it in various ways and check which matches one of the options provided.
Option 1: [tex]\( c = 10b - 10 - 5 \)[/tex]
This does not match our derived form because subtracting linear terms doesn't relate to squaring the expression.
Option 2: [tex]\( c = (10b - 10 - 5)^2 \)[/tex]
Simplifying inside the parentheses:
[tex]\[ 10b - 10 - 5 = 10b - 15 \][/tex]
Squaring it:
[tex]\[ c = (10b - 15)^2 \][/tex]
This form does not match [tex]\( c = (2b - 2)^2 \)[/tex].
Option 3: [tex]\( c = \frac{(10b - 2)^2}{25} \)[/tex]
Simplifying,
[tex]\[ (10b - 2)^2 = 100b^2 - 40b + 4 \][/tex]
Divide by 25:
[tex]\[ c = \frac{100b^2 - 40b + 4}{25} = 4b^2 - \frac{8b}{5} + \frac{4}{25} \][/tex]
This form does not match [tex]\( c = (2b - 2)^2 \)[/tex].
Option 4: [tex]\( c = \frac{(10b - 10)^2}{25} \)[/tex]
Simplifying inside the parentheses:
[tex]\[ 10b - 10 = 10(b - 1) \][/tex]
Squaring it:
[tex]\[ (10(b - 1))^2 = 100(b - 1)^2 \][/tex]
Divide by 25:
[tex]\[ c = \frac{100(b - 1)^2}{25} = 4(b - 1)^2 \][/tex]
Since [tex]\( 4(b - 1)^2 = (2b - 2)^2 \)[/tex],
[tex]\[ c = 4(b - 1)^2 \][/tex]
This form matches our original equation.
Therefore, the correct equivalent equation is:
[tex]\[ c = \frac{(10b - 10)^2}{25} \][/tex]
