The equation provided is $x^2 = 45$, where $x$ is a real number. To solve for $x$, follow these steps:

Solution:

1. Start with the equation: $x^2 = 45$

2. Take the square root of both sides to solve for $x$: $x = \pm \sqrt{45}$

3. Simplify $\sqrt{45}$. Since $45 = 9 \times 5$, we can factor out the square root of 9: $x = \pm \sqrt{9 \times 5} = \pm \sqrt{9} \times \sqrt{5} = \pm 3\sqrt{5}$

Thus, the simplified solution is: $x = \pm 3\sqrt{5}$

Final Answer:

$x = 3\sqrt{5} \quad \text{or} \quad x = -3\sqrt{5}$

Would you like further details or clarification?

Here are some related questions for further practice:

1. What is the general method for solving quadratic equations?
2. How do you simplify square roots involving products, like $\sqrt{45}$?
3. What are the properties of real numbers relevant to solving quadratic equations?
4. How would the solution change if the equation were $x^2 = -45$?
5. What if $x$ were required to be a positive number only?

Tip: Always check if the number under a square root can be simplified by factoring it into a perfect square times another number.