The given equation is:

$x^2 = -81$

Step 1: Analyze the Equation

To solve using the square root property, we need to take the square root of both sides. However, the equation has a negative value on the right-hand side, indicating that the solutions will involve complex numbers.

Step 2: Solve for $x$

$x^2 = -81$

Take the square root of both sides: $x = \pm \sqrt{-81}$

The square root of a negative number introduces the imaginary unit $i$, where $i = \sqrt{-1}$. Thus: $x = \pm \sqrt{81} \cdot \sqrt{-1}$ $x = \pm 9i$

Final Answer:

The solution set is: $\{9i, -9i\}$

Let me know if you want a detailed explanation or have any questions!

Here are 5 related questions to expand on this topic:

1. How do we interpret the imaginary unit $i$ in mathematics?
2. What are the properties of square roots when dealing with negative numbers?
3. Can all quadratic equations have complex solutions? Why or why not?
4. How would the solution change if the equation was $x^2 = 81$?
5. How do we plot complex numbers like $9i$ and $-9i$ on the complex plane?

Tip:

When solving equations involving imaginary numbers, always simplify using $i = \sqrt{-1}$ before completing the operations.