To find the solutions for the equation [tex]\((x - 9)^2 = 25\)[/tex], we will go through the following steps:
1. Understand the equation:
The given equation is [tex]\((x - 9)^2 = 25\)[/tex]. This is a quadratic equation where a square of the binomial [tex]\((x - 9)\)[/tex] is equal to 25.
2. Taking the square root of both sides:
To solve for [tex]\(x\)[/tex], we need to eliminate the square by taking the square root of both sides of the equation:
[tex]\[
\sqrt{(x - 9)^2} = \sqrt{25}
\][/tex]
The square root of a squared term gives us two possible solutions because [tex]\(\sqrt{a^2} = \pm a\)[/tex]:
[tex]\[
x - 9 = \pm 5
\][/tex]
3. Breaking it into two separate equations:
This can be split into two linear equations:
[tex]\[
x - 9 = 5 \quad \text{and} \quad x - 9 = -5
\][/tex]
4. Solving the first equation:
Solve for [tex]\(x\)[/tex] when [tex]\(x - 9 = 5\)[/tex]:
[tex]\[
x - 9 = 5
\][/tex]
[tex]\[
x = 5 + 9
\][/tex]
[tex]\[
x = 14
\][/tex]
5. Solving the second equation:
Solve for [tex]\(x\)[/tex] when [tex]\(x - 9 = -5\)[/tex]:
[tex]\[
x - 9 = -5
\][/tex]
[tex]\[
x = -5 + 9
\][/tex]
[tex]\[
x = 4
\][/tex]
Thus, the solutions to the equation [tex]\((x - 9)^2 = 25\)[/tex] are:
[tex]\[
x = 14 \quad \text{and} \quad x = 4
\][/tex]
6. Selecting the valid options:
From the given choices, we need to select those that match our solutions:
- 14
- [tex]$-14$[/tex]
- [tex]$-8$[/tex]
- 8
- 4
- [tex]$-4$[/tex]
We found that the solutions are [tex]\(x = 14\)[/tex] and [tex]\(x = 4\)[/tex]. Therefore, the valid options are:
[tex]\[
14 \quad \text{and} \quad 4
\][/tex]
So, the correct options are:
[tex]\[
\boxed{14 \quad \text{and} \quad 4}
\][/tex]
