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