The given equation is:

$x^2 - 81 = 0$

We will solve this algebraically.

Step 1: Rearrange the equation

$x^2 = 81$

Step 2: Solve for $x$

Taking the square root of both sides:

$x = \pm \sqrt{81}$ $x = \pm 9$

Thus, the two solutions are $x_1 = -9$ and $x_2 = 9$, where $x_1 \leq x_2$.

Final Answer:

$x_1 = -9 \quad \text{and} \quad x_2 = 9$

Do you have any questions or need further clarification?

Here are 5 related questions to expand on this topic:

1. What happens if we have an equation like $x^2 - 100 = 0$?
2. How can we apply the quadratic formula to solve this equation?
3. What are the steps for solving higher degree polynomial equations?
4. What is the graphical interpretation of the solutions for $x^2 - 81 = 0$?
5. Can this equation be solved using factoring?

Tip: When solving quadratic equations, always check if you can factor them easily before using more complex methods like the quadratic formula.