When [tex]$10b = 5(\sqrt{c} + 2)$[/tex] is solved for [tex]$c$[/tex], one equation is [tex]$c = (2b - 2)^2$[/tex].

Which of the following is an equivalent equation to find [tex][tex]$c$[/tex][/tex]?

A. [tex]c = 10b - 10 - 5[/tex]

B. [tex]c = (10b - 10 - 5)^2[/tex]

C. [tex]c = \frac{(10b - 2)^2}{25}[/tex]

D. [tex]c = \frac{(10b - 10)^2}{25}[/tex]



Answer :

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]