Answered

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When [tex][tex]$10b = 5(\sqrt{c} + 2)$[/tex][/tex] is solved for [tex][tex]$c$[/tex][/tex], one equation is [tex][tex]$c = (2b - 2)^2$[/tex][/tex]. Which of the following is an equivalent equation to find [tex][tex]$c$[/tex][/tex]?

A. [tex][tex]$c = 10b - 10 - 5$[/tex][/tex]
B. [tex][tex]$c = (10b - 10 - 5)^2$[/tex][/tex]
C. [tex][tex]$c = \frac{(10b - 2)^2}{25}$[/tex][/tex]
D. [tex][tex]$c = \frac{(10b - 10)^2}{25}$[/tex][/tex]

Sagot :

GinnyAnswer
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]
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