We are given the equation $x^2 = -121$ and we need to solve for $x$.

1. Problem Description

We are given the equation $x^2 = -121$ and we need to solve for $x$.

2. Solution Steps

To solve for $x$, we need to take the square root of both sides of the equation:

$x^2 = -121$

Taking the square root of both sides gives:

$x = \pm \sqrt{-121}$

Since the square root of a negative number is an imaginary number, we can rewrite the equation as:

$x = \pm \sqrt{121 \cdot (-1)}$

We know that $\sqrt{121} = 11$ and $\sqrt{-1} = i$, where $i$ is the imaginary unit. Therefore,

$x = \pm 11i$

3. Final Answer

The solution is $x = \pm 11i$.