Question

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

Answer 1

To solve the equation \(- (x - 3)^2 - 3 = 7\), we start by isolating the quadratic expression. First, add 3 to both sides of the equation:

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

Next, multiply both sides by -1:

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

Now we take the square root of both sides. Remember that the square root of a negative number will involve imaginary numbers:

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

This can be rewritten using the imaginary unit \(i\):

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

Now, we can solve for \(x\) by adding 3 to both sides:

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

Thus, the solutions to the equation are:

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

These can be expressed in the standard form of a complex number \(a + bi\):

1. \(x = 3 + \sqrt{10} i\)
2. \(x = 3 - \sqrt{10} i\)

Hence, the solutions in standard form are:

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