To solve the equation [tex]\((x-4)^2 = 5\)[/tex], let's break it down step-by-step:
1. Understand the Equation: We start with [tex]\((x-4)^2 = 5\)[/tex]. This is a quadratic equation in a form that suggests taking the square root of both sides.
2. Taking Square Roots: Taking the square root of both sides gives us two possible equations because both the positive and negative roots need to be considered:
[tex]\[
\sqrt{(x-4)^2} = \sqrt{5}
\][/tex]
which simplifies to:
[tex]\[
|x-4| = \sqrt{5}
\][/tex]
3. Splitting into Two Equations: The absolute value equation [tex]\(|x-4| = \sqrt{5}\)[/tex] means:
[tex]\[
x-4 = \sqrt{5} \quad \text{or} \quad x-4 = -\sqrt{5}
\][/tex]
4. Solving Each Equation:
- From [tex]\(x-4 = \sqrt{5}\)[/tex]:
[tex]\[
x = 4 + \sqrt{5}
\][/tex]
- From [tex]\(x-4 = -\sqrt{5}\)[/tex]:
[tex]\[
x = 4 - \sqrt{5}
\][/tex]
5. List the Solutions: The solutions to the equation [tex]\((x-4)^2 = 5\)[/tex] are:
[tex]\[
x = 4 + \sqrt{5} \quad \text{and} \quad x = 4 - \sqrt{5}
\][/tex]
6. Matching with Provided Choices: We now match our solutions with the given choices:
- A. [tex]\(x=9\)[/tex] and [tex]\(x=-1\)[/tex] - This does not match.
- B. [tex]\(x=4 \pm \sqrt{5}\)[/tex] - This matches our solutions exactly.
- C. [tex]\(x=-4 \pm \sqrt{5}\)[/tex] - This does not match.
- D. [tex]\(x=5 \pm \sqrt{4}\)[/tex] - This does not match.
Therefore, the correct choice is:
[tex]\[ \boxed{B} \][/tex]
