To solve the equation $x^2 + 144 = 0$, follow these steps:

1. Subtract 144 from both sides: $x^2 = -144$

2. Take the square root of both sides: $x = \pm \sqrt{-144}$

3. Since the square root of a negative number involves an imaginary unit ($i$), we have: $x = \pm \sqrt{144}i$

4. Simplifying the square root: $x = \pm 12i$

Thus, the solution is: $x = \pm 12i$

Let me know if you'd like more details or have questions!

Here are some related questions for further exploration:

1. What is the general form of a quadratic equation with imaginary solutions?
2. How do you solve other quadratic equations involving imaginary numbers?
3. What is the difference between real and imaginary solutions to quadratic equations?
4. How are imaginary numbers applied in engineering or physics?
5. Can a quadratic equation have both real and imaginary solutions?

Tip: Imaginary numbers arise from square roots of negative numbers and are expressed as multiples of $i$, where $i = \sqrt{-1}$.