Question

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

Answer 1

To solve the equation

\[
-(x - 3)^2 - 3 = 7,
\]

we first isolate the squared term. We can add 3 to both sides:

\[
-(x - 3)^2 = 7 + 3,
\]
\[
-(x - 3)^2 = 10.
\]

Next, we can multiply both sides by -1:

\[
(x - 3)^2 = -10.
\]

Now, we take the square root of both sides. Remember that taking the square root of a negative number involves using imaginary numbers:

\[
x - 3 = \pm \sqrt{-10}.
\]

We can express \(\sqrt{-10}\) using imaginary numbers:

\[
\sqrt{-10} = \sqrt{10}i.
\]

Thus, we have:

\[
x - 3 = \pm \sqrt{10} i.
\]

To solve for \(x\), we add 3 to both sides:

\[
x = 3 \pm \sqrt{10}i.
\]

Now we can express the solutions in standard complex number form \(a \pm bi\):

\[
x = 3 + \sqrt{10}i \quad \text{and} \quad x = 3 - \sqrt{10}i.
\]

Thus, the solutions are

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