To solve the equation \(-(x-3)^2 - 3 = 7\), we will first simplify and rearrange the equation.
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Start by adding 3 to both sides: \[ -(x - 3)^2 = 7 + 3 \] \[ -(x - 3)^2 = 10 \]
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Multiply both sides by -1: \[ (x - 3)^2 = -10 \]
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To solve for \(x - 3\), take the square root of both sides: \[ x - 3 = \pm \sqrt{-10} \]
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Recall that \(\sqrt{-10} = \sqrt{10}i\) (where \(i\) is the imaginary unit): \[ x - 3 = \pm \sqrt{10} i \]
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Now solve for \(x\): \[ x = 3 \pm \sqrt{10} i \]
So the solutions are: \[ x = 3 + \sqrt{10} i \quad \text{and} \quad x = 3 - \sqrt{10} i \]
In complex number standard form \(a + bi\), we can express the answer as: \[ 3 \pm i\sqrt{10} \]
Looking at the options given:
A: \(-7\)
B: \(-3 \pm i\sqrt{10}\)
C: \(3 \pm i\sqrt{10}\)
D: \(3 \pm 10i\)
The correct answer is:
C: \(3 \pm i\sqrt{10}\)
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