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.