Question

solve the rational equation. remember to check for extraneous solutions.
$\frac{p + 4}{4p^2} = \frac{1}{2p^2} + \frac{p - 4}{2p^2}$
a {10}
b {-5}
c {-10, -5}
d {-3}

Explanation:

Step1: Find common denominator

The equation is \(\frac{p + 4}{4p^2}=\frac{1}{2p^2}+\frac{p - 4}{2p^2}\). The common denominator of the right - hand side is \(2p^2\), and the left - hand side denominator is \(4p^2\). Multiply the numerator and denominator of the fractions with denominator \(2p^2\) by 2 to get a common denominator of \(4p^2\). So \(\frac{1}{2p^2}=\frac{2}{4p^2}\) and \(\frac{p - 4}{2p^2}=\frac{2(p - 4)}{4p^2}\). The equation becomes \(\frac{p + 4}{4p^2}=\frac{2}{4p^2}+\frac{2(p - 4)}{4p^2}\)

Step2: Eliminate denominators

Since the denominators are the same (\(p
eq0\) to avoid division by zero), we can set the numerators equal:
\(p + 4=2+2(p - 4)\)

Step3: Expand and simplify

Expand the right - hand side: \(p + 4=2 + 2p-8\)
Simplify the right - hand side: \(p + 4=2p - 6\)
Subtract \(p\) from both sides: \(4=p - 6\)
Add 6 to both sides: \(p=10\)

Step4: Check for extraneous solutions

Substitute \(p = 10\) into the original equation:
Left - hand side: \(\frac{10 + 4}{4\times10^2}=\frac{14}{400}=\frac{7}{200}\)
Right - hand side: \(\frac{1}{2\times10^2}+\frac{10 - 4}{2\times10^2}=\frac{1}{200}+\frac{6}{200}=\frac{7}{200}\)
Since \(p = 10\) does not make any denominator zero, it is a valid solution.

Answer:

A. \(\{10\}\)