To solve the quadratic equation \( x^2 + 16 = 0 \), we start by isolating \( x^2 \):
\[ x^2 = -16 \]
Next, we take the square root of both sides. Since we are dealing with a negative number, we will introduce the imaginary unit \( i \), where \( i = \sqrt{-1} \):
\[ x = \pm \sqrt{-16} \]
This can be rewritten as:
\[ x = \pm \sqrt{16} \cdot \sqrt{-1} = \pm 4i \]
Thus, the solutions to the equation \( x^2 + 16 = 0 \) are:
\[ x = 0 \pm 4i \]
This can be expressed in the form \( a \pm bi \) as:
\[ x = 0 \pm 4i \]
So, the final solutions are:
\[ x = 0 + 4i \quad \text{and} \quad x = 0 - 4i \]