To solve the equation \( \frac{1}{x} + \frac{6}{x^2} = 1 \), follow these steps:

### Step 1: Multiply the entire equation by \( x^2 \) to eliminate the denominators.
\[
x^2 \left(\frac{1}{x} + \frac{6}{x^2}\right) = x^2 \times 1
\]
This simplifies to:
\[
x \cdot x + 6 = x^2
\]

### Step 2: Simplify the equation.
\[
x + 6 = x^2
\]

### Step 3: Rearrange the equation to form a standard quadratic equation.
\[
x^2 - x - 6 = 0
\]

### Step 4: Factor the quadratic equation.
We need to find two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\). So, we can factor the equation as:
\[
(x - 3)(x + 2) = 0
\]

### Step 5: Solve for \( x \).
Set each factor equal to zero:
\[
x - 3 = 0 \quad \text{or} \quad x + 2 = 0
\]
\[
x = 3 \quad \text{or} \quad x = -2
\]

### Final Answer:
The solutions are \( x = 3 \) and \( x = -2 \).

Would you like further details on this solution or have any questions?

Here are five related questions:
1. How do you solve a quadratic equation using the quadratic formula?
2. What is the process of completing the square to solve a quadratic equation?
3. How can you graph a quadratic equation to find its solutions?
4. What are the properties of the discriminant in a quadratic equation?
5. Can a quadratic equation have complex solutions?

**Tip:** Always check your solutions by substituting them back into the original equation to verify they satisfy it.

Explore how to solve the equation 1/x + 6/x² = 1 by transforming it into a quadratic equation. This detailed guide covers essential algebraic concepts and factoring techniques, ideal for students in Grades 8-10.

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