To find the factorization of the quadratic expression [tex]\(x^2 - 7x + 12\)[/tex], let's follow a step-by-step process:
### Step 1: Identify the general form of a quadratic expression
A quadratic expression can generally be written as:
[tex]\[ ax^2 + bx + c \][/tex]
For the given quadratic expression [tex]\(x^2 - 7x + 12\)[/tex]:
- The coefficient [tex]\( a \)[/tex] is 1.
- The coefficient [tex]\( b \)[/tex] is -7.
- The constant term [tex]\( c \)[/tex] is 12.
### Step 2: Find the roots of the quadratic equation using the quadratic formula
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the coefficients from the quadratic expression.
Substituting the values [tex]\( a = 1 \)[/tex], [tex]\( b = -7 \)[/tex], and [tex]\( c = 12 \)[/tex]:
1. Calculate the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac \][/tex]
[tex]\[ \text{Discriminant} = (-7)^2 - 4 \cdot 1 \cdot 12 \][/tex]
[tex]\[ \text{Discriminant} = 49 - 48 \][/tex]
[tex]\[ \text{Discriminant} = 1 \][/tex]
2. Calculate the roots:
[tex]\[ x_1 = \frac{-b + \sqrt{\text{Discriminant}}}{2a} \][/tex]
[tex]\[ x_1 = \frac{7 + \sqrt{1}}{2 \cdot 1} \][/tex]
[tex]\[ x_1 = \frac{7 + 1}{2} \][/tex]
[tex]\[ x_1 = \frac{8}{2} \][/tex]
[tex]\[ x_1 = 4 \][/tex]
[tex]\[ x_2 = \frac{-b - \sqrt{\text{Discriminant}}}{2a} \][/tex]
[tex]\[ x_2 = \frac{7 - \sqrt{1}}{2 \cdot 1} \][/tex]
[tex]\[ x_2 = \frac{7 - 1}{2} \][/tex]
[tex]\[ x_2 = \frac{6}{2} \][/tex]
[tex]\[ x_2 = 3 \][/tex]
The roots of the quadratic equation [tex]\(x^2 - 7x + 12 = 0\)[/tex] are [tex]\(x = 4\)[/tex] and [tex]\(x = 3\)[/tex].
### Step 3: Write the factorization
Given the roots [tex]\(x = 4\)[/tex] and [tex]\(x = 3\)[/tex], the quadratic expression can be factored into:
[tex]\[ (x - 4)(x - 3) \][/tex]
### Step 4: Match the factorization with the given options
Compare the factorization [tex]\((x - 4)(x - 3)\)[/tex] with the provided multiple-choice options:
A. [tex]\((x+6)(x+8)\)[/tex]
B. [tex]\((x-3)(x-4)\)[/tex]
C. [tex]\((x+3)(x+4)\)[/tex]
D. [tex]\((x-6)(x-8)\)[/tex]
Option B: [tex]\((x - 3)(x - 4)\)[/tex] is equivalent to [tex]\((x - 4)(x - 3)\)[/tex] (since multiplication is commutative).
Therefore, the correct answer is:
[tex]\[ \boxed{(x-3)(x-4)} \][/tex]
So, the answer is Option B.
