Question

Solve the equation −(x−3)2−3=7
and re-express the answer as a complex number in the standard form a±bi
.(1 point)

Answer 1

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:

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