Sure, let's go through the solution step-by-step.
Given the equation:
[tex]\[
(x - 8)^2 = -16
\][/tex]
I will show you how to solve it step-by-step.
1. Recognize the given quadratic form:
[tex]\[
(x - 8)^2 = -16
\][/tex]
This is a standard form where we have a square of a binomial on the left-hand side.
2. Take the square root of both sides:
By taking the square root of both sides, we get:
[tex]\[
\sqrt{(x - 8)^2} = \sqrt{-16}
\][/tex]
On the left side, the square root of the square cancels out, leaving:
[tex]\[
x - 8 = \pm \sqrt{-16}
\][/tex]
3. Simplify the right-hand side:
We know that [tex]\(\sqrt{-16}\)[/tex] involves an imaginary number because we are taking the square root of a negative number.
[tex]\(\sqrt{-16}\)[/tex] can be rewritten using [tex]\(i\)[/tex], the imaginary unit ([tex]\(i = \sqrt{-1}\)[/tex]):
[tex]\[
\sqrt{-16} = 4i
\][/tex]
So, our equation now looks like:
[tex]\[
x - 8 = \pm 4i
\][/tex]
4. Separate the equation into two cases:
We have two potential solutions based on the [tex]\(\pm\)[/tex]:
[tex]\[
x - 8 = 4i \quad \text{and} \quad x - 8 = -4i
\][/tex]
5. Solve for [tex]\(x\)[/tex] in each case:
For the first case:
[tex]\[
x - 8 = 4i
\][/tex]
Adding 8 to both sides:
[tex]\[
x = 8 + 4i
\][/tex]
For the second case:
[tex]\[
x - 8 = -4i
\][/tex]
Adding 8 to both sides:
[tex]\[
x = 8 - 4i
\][/tex]
6. Write the final solutions:
Thus, the solutions to the original equation are:
[tex]\[
x_1 = 8 + 4i
\][/tex]
[tex]\[
x_2 = 8 - 4i
\][/tex]
These are the complex solutions to the quadratic equation [tex]\((x - 8)^2 = -16\)[/tex].
