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Answer :
To factor the expression [tex]\(\frac{1}{25} p^2 - 100\)[/tex], we can use the difference of squares formula. A difference of squares is an expression in the form [tex]\(a^2 - b^2\)[/tex], and it factors into [tex]\((a + b)(a - b)\)[/tex].
Here's how we can apply it step-by-step:
1. Identify the square terms:
The given expression is [tex]\(\frac{1}{25} p^2 - 100\)[/tex].
Let's rewrite [tex]\(\frac{1}{25} p^2\)[/tex] as [tex]\(\left(\frac{1}{5} p\right)^2\)[/tex].
Similarly, write [tex]\(100\)[/tex] as [tex]\(10^2\)[/tex].
2. Expression in terms of squares:
Now, the expression becomes [tex]\(\left(\frac{1}{5} p\right)^2 - 10^2\)[/tex].
3. Apply the difference of squares formula:
Now that we have the expression [tex]\(a^2 - b^2\)[/tex] where [tex]\(a = \frac{1}{5} p\)[/tex] and [tex]\(b = 10\)[/tex], we can factor it using:
[tex]\[
a^2 - b^2 = (a + b)(a - b)
\][/tex]
Substituting the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex], we get:
[tex]\[
\left(\frac{1}{5} p + 10\right)\left(\frac{1}{5} p - 10\right)
\][/tex]
So, the factored form of [tex]\(\frac{1}{25} p^2 - 100\)[/tex] is [tex]\(\left(\frac{1}{5} p + 10\right)\left(\frac{1}{5} p - 10\right)\)[/tex], which matches option [tex]\(c\)[/tex].
Here's how we can apply it step-by-step:
1. Identify the square terms:
The given expression is [tex]\(\frac{1}{25} p^2 - 100\)[/tex].
Let's rewrite [tex]\(\frac{1}{25} p^2\)[/tex] as [tex]\(\left(\frac{1}{5} p\right)^2\)[/tex].
Similarly, write [tex]\(100\)[/tex] as [tex]\(10^2\)[/tex].
2. Expression in terms of squares:
Now, the expression becomes [tex]\(\left(\frac{1}{5} p\right)^2 - 10^2\)[/tex].
3. Apply the difference of squares formula:
Now that we have the expression [tex]\(a^2 - b^2\)[/tex] where [tex]\(a = \frac{1}{5} p\)[/tex] and [tex]\(b = 10\)[/tex], we can factor it using:
[tex]\[
a^2 - b^2 = (a + b)(a - b)
\][/tex]
Substituting the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex], we get:
[tex]\[
\left(\frac{1}{5} p + 10\right)\left(\frac{1}{5} p - 10\right)
\][/tex]
So, the factored form of [tex]\(\frac{1}{25} p^2 - 100\)[/tex] is [tex]\(\left(\frac{1}{5} p + 10\right)\left(\frac{1}{5} p - 10\right)\)[/tex], which matches option [tex]\(c\)[/tex].
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