Absolutely, let's go through the process of factoring the quadratic expression [tex]\( 2x^2 - 7x + 6 \)[/tex] in a step-by-step manner.
### Step 1: Understand the Quadratic Expression
We have the quadratic expression in the form:
[tex]\[ 2x^2 - 7x + 6 \][/tex]
### Step 2: Identify a, b, and c
Identify the coefficients a, b, and c from the standard quadratic form [tex]\( ax^2 + bx + c \)[/tex]:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = -7 \)[/tex]
- [tex]\( c = 6 \)[/tex]
### Step 3: Factoring the Quadratic Expression
We need to find two binomials of the form [tex]\( (mx + n)(px + q) \)[/tex] such that:
[tex]\[ (mx + n)(px + q) = 2x^2 - 7x + 6 \][/tex]
To factor this, we look for pairs (n1, n2) such that:
1. The product of the first terms is [tex]\( 2x^2 \)[/tex]. This implies [tex]\( m \cdot p = 2 \)[/tex].
2. The product of the constant terms equals the free term [tex]\(\ c\)[/tex]. This implies [tex]\( n \cdot q = 6 \)[/tex].
3. The sum of the products of the outer and inner terms equals the middle term [tex]\(\ −7x \)[/tex]. This implies [tex]\( n \cdot p + m \cdot q = -7 \)[/tex].
### Step 4: Find Suitable Factors
We explore different factors of 6 that could guide us to the correct middle term condition:
- Consider the pair [tex]\((x - 2)\)[/tex] and [tex]\( (2x - 3) \)[/tex]:
Let's check:
1. Product of first terms: [tex]\( 1 \cdot 2x^2 = 2x^2 \)[/tex] (matches [tex]\(2x^2\)[/tex]).
2. Product of last terms: [tex]\( -2 \cdot - 3 = 6 \)[/tex] (matches [tex]\(6\)[/tex]).
3. The middle term generation:
[tex]\[ -2 \cdot 2x + (- 3) \cdot x = -4x - 3x = -7x \][/tex] (matches [tex]\(-7x\)[/tex]).
### Step 5: Write the Factored Form
Thus, we have found the correct binomials:
[tex]\[ (x - 2)(2x - 3) \][/tex]
To verify, we can expand [tex]\( (x - 2)(2x - 3) \)[/tex]:
[tex]\[ (x - 2)(2x - 3) \][/tex]
[tex]\[ = x \cdot 2x + x \cdot -3 - 2 \cdot 2x + (-2) \cdot -3 \][/tex]
[tex]\[ = 2x^2 - 3x - 4x + 6 \][/tex]
[tex]\[ = 2x^2 - 7x + 6 \][/tex]
Since the expanded form matches the original quadratic equation, our factorization is correct.
Thus, the factorization of [tex]\( 2x^2 - 7x + 6 \)[/tex] is:
[tex]\[ (x - 2)(2x - 3) \][/tex]
So, substituting these into the blank boxes in:
[tex]\[
([?]x-3)(x-\square)
\][/tex]
We get:
[tex]\[
(2x-3)(x-2)
\][/tex]
Hence, the factored form of the given expression is:
[tex]\[
(2x-3)(x-2)
\][/tex]
