To solve the equation \(- (x - 3)^2 - 3 = 7\), we start by simplifying it.
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First, we add 3 to both sides of the equation: \[
- (x - 3)^2 = 7 + 3 \] \[
- (x - 3)^2 = 10 \]
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Next, we multiply both sides by -1 to eliminate the negative: \[ (x - 3)^2 = -10 \]
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Now, we take the square root of both sides. Remember that the square root of a negative number involves imaginary numbers: \[ x - 3 = \pm \sqrt{-10} \] \[ x - 3 = \pm i \sqrt{10} \]
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Finally, we solve for \(x\) by adding 3 to both sides: \[ x = 3 \pm i \sqrt{10} \]
Now, we can express the answer in standard form \(a \pm bi\): \[ x = 3 + i \sqrt{10} \quad \text{or} \quad x = 3 - i \sqrt{10} \]
So the solutions are: \[ x = 3 + i \sqrt{10} \quad \text{and} \quad x = 3 - i \sqrt{10} \]