Solve the equation $-(x-3)^2-3=7$ and re-express the answer as a complex number in the standard form.

Question

A. $3 \pm i √(10)$

B. [tex]$-3 \pm i √(10)$[/tex]

C. $3 \pm 2i$

D. $-3 \pm 2i$

Answer 1

To solve the equation \(- (x-3)^2 - 3 = 7\), we start by rearranging the equation.

1. First, isolate the squared term: \[

   • (x-3)^2 - 3 = 7 \] Add 3 to both sides: \[
   • (x-3)^2 = 10 \] Multiply through by -1: \[ (x-3)^2 = -10 \]

2. Next, take the square root of both sides. Remember that taking the square root of a negative number introduces the imaginary unit \(i\): \[ x - 3 = \pm \sqrt{-10} = \pm \sqrt{10} i \]

3. 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: \[ \boxed{3 \pm i \sqrt{10}} \]