To solve the equation [tex]\((x - 7)^2 + 4 = 0\)[/tex], we follow these steps:
1. Start with the given equation:
[tex]\[
(x - 7)^2 + 4 = 0
\][/tex]
2. Isolate the squared term by subtracting 4 from both sides:
[tex]\[
(x - 7)^2 = -4
\][/tex]
3. Observe that [tex]\((x - 7)^2 = -4\)[/tex] is an equation involving a negative number on one side. This indicates the solutions will be complex numbers. The equation [tex]\((x - 7)^2 = -4\)[/tex] implies that [tex]\(x - 7\)[/tex] squared equals a negative number, which can't happen for any real number [tex]\(x\)[/tex], leading us to consider complex solutions.
4. Take the square root of both sides, remembering to include both the positive and negative roots:
[tex]\[
x - 7 = \pm \sqrt{-4}
\][/tex]
5. Simplify [tex]\(\sqrt{-4}\)[/tex] using imaginary units:
[tex]\[
\sqrt{-4} = 2i
\][/tex]
So, we have:
[tex]\[
x - 7 = \pm 2i
\][/tex]
6. Solve for [tex]\(x\)[/tex] by adding 7 to each side:
[tex]\[
x = 7 \pm 2i
\][/tex]
7. This gives us two solutions:
[tex]\[
x_1 = 7 - 2i \quad \text{(the lower solution)}
\][/tex]
[tex]\[
x_2 = 7 + 2i \quad \text{(the higher solution)}
\][/tex]
So, the final solutions are:
[tex]\[
x_1 = 7 - 2i
\][/tex]
[tex]\[
x_2 = 7 + 2i
\][/tex]
