Blake knows that one of the solutions to [tex]$x^2-6x+8=0[tex]$[/tex] is [tex]$[/tex]x=2$[/tex]. What is the other solution?

A. [tex]$x=-6$[/tex]
B. [tex]$x=-3$[/tex]
C. [tex]$x=4$[/tex]
D. [tex]$x=8$[/tex]



Answer :

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]