Let's solve the equation [tex]\((x - 4)^2 = 5\)[/tex].
### Step-by-Step Solution:
1. Start with the given equation:
[tex]\[
(x - 4)^2 = 5.
\][/tex]
2. Take the square root of both sides to remove the squared term:
[tex]\[
\sqrt{(x - 4)^2} = \sqrt{5}.
\][/tex]
Recall that taking the square root of both sides gives us two separate equations because of the property [tex]\(\sqrt{a^2} = \pm a\)[/tex].
3. Apply the [tex]\(\pm\)[/tex] (plus and minus) sign:
[tex]\[
x - 4 = \pm \sqrt{5}.
\][/tex]
4. Separate this into two individual equations:
[tex]\[
x - 4 = \sqrt{5},
\][/tex]
and
[tex]\[
x - 4 = -\sqrt{5}.
\][/tex]
5. Solve each equation for [tex]\(x\)[/tex]:
- For the first equation:
[tex]\[
x - 4 = \sqrt{5} \implies x = 4 + \sqrt{5}.
\][/tex]
- For the second equation:
[tex]\[
x - 4 = -\sqrt{5} \implies x = 4 - \sqrt{5}.
\][/tex]
6. Combine the solutions:
Thus, we get two solutions for [tex]\(x\)[/tex]:
[tex]\[
x = 4 + \sqrt{5} \quad \text{and} \quad x = 4 - \sqrt{5}.
\][/tex]
### Final Answer:
By comparing these solutions with the given choices:
- A. [tex]\(x = 4 \pm \sqrt{5}\)[/tex]
- B. [tex]\(x = 9\)[/tex] and [tex]\(x = -1\)[/tex]
- C. [tex]\(x = 5 \pm \sqrt{4}\)[/tex]
- D. [tex]\(x = -4 \pm \sqrt{5}\)[/tex]
The correct answer is:
[tex]\[
\boxed{A. \ x = 4 \pm \sqrt{5}}
\][/tex]
