https://file.aiquickdraw.com/mathbot/file/1727866094597uaianbry.jpg

The equation given in the image is:

\[
x^{-2} - x^{-1} - 12 = 0
\]

### Step-by-step solution:

1. **Substitute for simpler expressions**: Let \( y = x^{-1} \) (so \( x^{-2} = y^2 \)).

   The equation now becomes:
   \[
   y^2 - y - 12 = 0
   \]

2. **Solve the quadratic equation**: Use the quadratic formula to solve for \( y \):
   \[
   y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-12)}}{2(1)}
   \]
   Simplifying the expression:
   \[
   y = \frac{1 \pm \sqrt{1 + 48}}{2}
   \]
   \[
   y = \frac{1 \pm \sqrt{49}}{2}
   \]
   \[
   y = \frac{1 \pm 7}{2}
   \]
   So, the solutions for \( y \) are:
   \[
   y = \frac{1 + 7}{2} = 4 \quad \text{or} \quad y = \frac{1 - 7}{2} = -3
   \]

3. **Return to \( x \)**: Recall that \( y = x^{-1} \), so \( x = \frac{1}{y} \).

   - If \( y = 4 \), then \( x = \frac{1}{4} \).
   - If \( y = -3 \), then \( x = \frac{1}{-3} = -\frac{1}{3} \).

### Final solutions:

The solutions for \( x \) are:
\[
x = \frac{1}{4} \quad \text{or} \quad x = -\frac{1}{3}
\]

---

Would you like further details or have any other questions?

### Related Questions:
1. How can we verify the solutions by substituting back into the original equation?
2. What would happen if we had a different constant instead of \( -12 \) in the equation?
3. How would the method change if there were higher powers of \( x^{-1} \) in the equation?
4. Can we solve this equation graphically? What would the graph of the equation look like?
5. How can this method be applied to other similar inverse power equations?

### Tip:
Always try to simplify complex-looking equations by substituting variables to make them easier to manage, especially when working with inverse powers.

Solve the equation: x^(-2) - x^(-1) - 12 = 0

Learn how to solve the inverse power equation x^(-2) - x^(-1) - 12 with a detailed step-by-step solution. This problem involves key algebraic concepts such as inverse powers, substitution, and the quadratic formula. Suitable for students in Grades 9-11.

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