To solve the equation 

\[
\frac{1}{x} + \frac{6}{x^2} = 1,
\]

let's follow these steps:

### Step 1: Find a common denominator
The common denominator is \(x^2\). Rewriting the terms over this common denominator gives:

\[
\frac{x + 6}{x^2} = 1.
\]

### Step 2: Clear the fraction by multiplying by \(x^2\)
Multiply both sides of the equation by \(x^2\):

\[
x + 6 = x^2.
\]

### Step 3: Rearrange the equation to standard quadratic form
Move all terms to one side to form a quadratic equation:

\[
x^2 - x - 6 = 0.
\]

### Step 4: Factor the quadratic equation
To factor \(x^2 - x - 6 = 0\), look for two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\). So, the equation factors as:

\[
(x - 3)(x + 2) = 0.
\]

### Step 5: Solve for \(x\)
Set each factor equal to zero:

\[
x - 3 = 0 \quad \text{or} \quad x + 2 = 0.
\]

So, the solutions are:

\[
x = 3 \quad \text{or} \quad x = -2.
\]

Thus, the values of \(x\) that satisfy the equation are \(x = 3\) and \(x = -2\).

Would you like more details or have any questions?

Here are some related questions you might explore:
1. How do you solve similar equations with different coefficients?
2. What happens if there is no real solution for a quadratic equation?
3. How do you complete the square to solve a quadratic equation?
4. How can quadratic equations be solved using the quadratic formula?
5. What are the applications of solving rational equations in real life?

**Tip:** Always check your solutions by substituting them back into the original equation to ensure they are correct!

Learn how to solve the equation 1/x + 6/x² = 1 with detailed step-by-step guidance. This solution covers key algebraic concepts, including rational equations, converting to a quadratic form, and factoring. Suitable for students in Grades 8-10.

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