Solve the equation −(x−3)2−3=7

Question

−
(
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

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