To solve for the other root of the quadratic equation \( x^2 - 6x + 8 = 0 \), we can start by using the information given that one of the solutions is \( x = 2 \).
### Step-by-Step Solution:
1. Identify the given root: We are given that \( x = 2 \) is one of the solutions.
2. Factorize the quadratic equation: Knowing one solution helps us factorize the quadratic equation. The quadratic equation can be expressed as:
[tex]\[
x^2 - 6x + 8 = (x - 2)(x - b) = 0
\][/tex]
Here, \( x = 2 \) is one factor, so one part of our factorized form is \( (x - 2) \).
3. Find the second factor: Next, we need to determine the other factor. Since we have \( x^2 - 6x + 8 \), which can be rewritten as:
[tex]\[
x^2 - 6x + 8 = (x - 2)(x - b)
\][/tex]
where \( b \) is the other root.
4. Expanding and comparing coefficients:
[tex]\[
(x - 2)(x - b) = x^2 - (2 + b)x + 2b
\][/tex]
By comparing this with the original equation \( x^2 - 6x + 8 \), we get:
[tex]\[
2 + b = 6 \quad \text{and} \quad 2b = 8
\][/tex]
5. Solving these equations:
- For the first equation: \( 2 + b = 6 \)
[tex]\[
b = 6 - 2
\][/tex]
[tex]\[
b = 4
\][/tex]
- For the second equation: \( 2b = 8 \)
[tex]\[
b = \frac{8}{2}
\][/tex]
[tex]\[
b = 4
\][/tex]
Both methods agree that \( b = 4 \).
Therefore, the other solution to the equation \( x^2 - 6x + 8 = 0 \) is \( x = 4 \).
Thus, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
