To determine which of the given options is equivalent to the quadratic expression [tex]\(x^2 - x - 6\)[/tex], we need to factor the quadratic expression properly.
We begin with the expression:
[tex]\[ x^2 - x - 6 \][/tex]
### Step-by-Step Solution:
1. Identify the factors:
For a quadratic expression of the form [tex]\(ax^2 + bx + c\)[/tex], we aim to find two numbers whose product is [tex]\(ac\)[/tex] (which is the product of the coefficient of [tex]\(x^2\)[/tex] and the constant term, [tex]\(a \cdot c\)[/tex]) and whose sum is [tex]\(b\)[/tex] (the coefficient of [tex]\(x\)[/tex]).
In this case, [tex]\(a = 1\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(c = -6\)[/tex].
- Product (ac): [tex]\(1 \cdot (-6) = -6\)[/tex]
- Sum (b): [tex]\(-1\)[/tex]
2. Find pairs of numbers:
We need to find two numbers that multiply to [tex]\(-6\)[/tex] and add up to [tex]\(-1\)[/tex]. The pairs of numbers that multiply to [tex]\(-6\)[/tex] are:
- [tex]\(1\)[/tex] and [tex]\(-6\)[/tex]
- [tex]\(-1\)[/tex] and [tex]\(6\)[/tex]
- [tex]\(2\)[/tex] and [tex]\(-3\)[/tex]
- [tex]\(-2\)[/tex] and [tex]\(3\)[/tex]
The pair that adds up to [tex]\(-1\)[/tex] is [tex]\(2\)[/tex] and [tex]\(-3\)[/tex].
3. Rewrite and factor:
Rewriting [tex]\(x^2 - x - 6\)[/tex] using [tex]\(2\)[/tex] and [tex]\(-3\)[/tex]:
[tex]\[ x^2 - x - 6 = x^2 + 2x - 3x - 6 \][/tex]
Grouping terms and factoring by grouping:
[tex]\[ x^2 + 2x - 3x - 6 = x(x + 2) - 3(x + 2) \][/tex]
[tex]\[ = (x - 3)(x + 2) \][/tex]
4. Match with the options:
Reviewing the options provided:
- A [tex]\(\quad(x+2)(x-3)\)[/tex] matches [tex]\((x - 3)(x + 2)\)[/tex]
- B [tex]\(\quad(x-2)(x+3)\)[/tex]
- C [tex]\((x-2)(x-3)\)[/tex]
- D [tex]\(\quad(x-6)(x+1)\)[/tex]
- E [tex]\(\quad(x+6)(x-1)\)[/tex]
Therefore, the equivalent expression for [tex]\(x^2 - x - 6\)[/tex] is:
[tex]\[ \boxed{A \quad (x + 2)(x - 3)} \][/tex]
