https://file.aiquickdraw.com/mathbot/file/172736056084820ttwco2.jpg

To solve the equation given in the image:

\[
\frac{3}{2p^2} + \frac{5}{4p^2} = \frac{1}{4p}
\]

we will find a common denominator and then solve for \( p \).

### Step 1: Combine the fractions on the left-hand side
The common denominator for the fractions \(\frac{3}{2p^2}\) and \(\frac{5}{4p^2}\) is \(4p^2\). Rewrite the fractions with this common denominator:

\[
\frac{3}{2p^2} = \frac{3 \times 2}{2p^2 \times 2} = \frac{6}{4p^2}
\]

\[
\frac{5}{4p^2} = \frac{5}{4p^2}
\]

Now combine them:

\[
\frac{6}{4p^2} + \frac{5}{4p^2} = \frac{6 + 5}{4p^2} = \frac{11}{4p^2}
\]

### Step 2: Set up the equation
Now substitute this into the original equation:

\[
\frac{11}{4p^2} = \frac{1}{4p}
\]

### Step 3: Cross-multiply
Cross-multiply to eliminate the fractions:

\[
11 \cdot 4p = 1 \cdot 4p^2
\]

This simplifies to:

\[
44p = 4p^2
\]

### Step 4: Simplify the equation
Divide both sides by 4:

\[
11p = p^2
\]

Rearrange to form a quadratic equation:

\[
p^2 - 11p = 0
\]

### Step 5: Factor the quadratic equation
Factor out \( p \):

\[
p(p - 11) = 0
\]

So, the solutions are:

\[
p = 0 \quad \text{or} \quad p = 11
\]

Since \( p = 0 \) would make the original fractions undefined (division by zero), the valid solution is:

\[
\boxed{p = 11}
\]

Thus, the correct answer is **C. \( p = 11 \)**.

Would you like further details or have any questions?

### Related Questions:
1. How do you solve equations with fractions and variable terms in the denominator?
2. What is cross-multiplication, and when is it used in solving equations?
3. How can factoring help solve quadratic equations?
4. Why does the solution \( p = 0 \) not work in this equation?
5. How would this process change if there were more complex terms in the fractions?

### Tip:
Always check for undefined values when solving equations with variables in the denominator!

Solve for p in the equation \(\frac{3}{2p^2} + \frac{5}{4p^2} = \frac{1}{4p}\).

Learn how to solve the fractional equation \(\frac{3}{2p^2} + \frac{5}{4p^2} = \frac{1}{4p}\) step-by-step. This problem involves finding a common denominator, using cross-multiplication, and solving the resulting quadratic equation. Detailed explanation suitable for high school students in Grades 9-12.

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