To solve the equation:

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

Let's go through the steps:

### Step 1: Combine terms on the left-hand side
Rewrite the equation to have a common denominator on the left-hand side.

\[
\frac{4}{x} + \frac{1}{x^2} = \frac{4x}{x^2} + \frac{1}{x^2}
\]

Now, combine the fractions:

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

### Step 2: Eliminate the denominator
Multiply both sides of the equation by \(x^2\) to eliminate the denominators:

\[
4x + 1 = \frac{1}{5}
\]

### Step 3: Isolate \(x\)
Now, multiply through by 5 to clear the fraction:

\[
5(4x + 1) = 1
\]

This simplifies to:

\[
20x + 5 = 1
\]

### Step 4: Solve for \(x\)
Subtract 5 from both sides:

\[
20x = -4
\]

Finally, divide by 20:

\[
x = -\frac{4}{20} = -\frac{1}{5}
\]

Thus, the solution is:

\[
x = -\frac{1}{5}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you solve similar rational equations?
2. What are the different methods to simplify fractions?
3. What happens when variables appear in both the numerator and denominator?
4. How do you solve quadratic equations involving fractions?
5. Can such equations have more than one solution?

**Tip:** Always check for extraneous solutions, especially when multiplying both sides of an equation by variables.

Explore a step-by-step solution to solve the rational equation 4/x + 1/x^2 = 1/5x^2. Learn key concepts in algebra and fraction manipulation, making it ideal for students in Grades 8-10. This guide simplifies the process of working with fractions in equations.

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