To solve the equation \(- (x-3)^2 - 3 = 7\), we start by rearranging the equation.
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First, isolate the squared term: \[
- (x-3)^2 - 3 = 7 \] Add 3 to both sides: \[
- (x-3)^2 = 10 \] Multiply through by -1: \[ (x-3)^2 = -10 \]
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Next, take the square root of both sides. Remember that taking the square root of a negative number introduces the imaginary unit \(i\): \[ x - 3 = \pm \sqrt{-10} = \pm \sqrt{10} i \]
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Now, solve for \(x\) by adding 3 to both sides: \[ x = 3 \pm \sqrt{10} i \]
The solutions are expressed as: \[ x = 3 + \sqrt{10} i \quad \text{and} \quad x = 3 - \sqrt{10} i \]
In standard form for complex numbers, this can be written as: \[ x = 3 \pm i \sqrt{10} \]
Thus, the answer is option A: \[ \boxed{3 \pm i \sqrt{10}} \]