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To solve the equation \( x^2 + 9 = 45 \), we start by isolating \( x^2 \):
\[
x^2 + 9 = 45
\]
Subtracting 9 from both sides:
\[
x^2 = 45 - 9
\]
\[
x^2 = 36
\]
Next, we take the square root of both sides to find \( x \):
\[
x = \pm \sqrt{36}
\]
\[
x = \pm 6
\]
This gives us the solutions \( x = 6 \) and \( x = -6 \). Thus, the solution set is:
\[
\{-6, 6\}
\]
Amelia is correct in stating that the solution set is \(\{-6, 6\}\). Therefore, the answer is:
**Amelia**.