To solve the equation \(- (x - 3)^2 - 3 = 7\), we can first isolate the squared term. Let's start by moving -3 to the other side:
\[ -(x - 3)^2 = 7 + 3 \]
This simplifies to:
\[ -(x - 3)^2 = 10 \]
Now, multiply both sides by -1:
\[ (x - 3)^2 = -10 \]
Next, we take the square root of both sides. Remember that the square root of a negative number involves the imaginary unit \(i\):
\[ x - 3 = \pm \sqrt{-10} \]
This can be expressed as:
\[ x - 3 = \pm i \sqrt{10} \]
Now, isolate \(x\):
\[ x = 3 \pm i \sqrt{10} \]
Thus, we have two solutions:
\[ x = 3 + i\sqrt{10} \quad \text{and} \quad x = 3 - i\sqrt{10} \]
In standard form \(a \pm bi\), the answer can be expressed as:
\[ \boxed{3 \pm i\sqrt{10}} \]
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