To factor the quadratic expression [tex]\(x^2 + 4x - 12\)[/tex], follow these steps:
1. Identify the coefficients: The quadratic expression is of the form [tex]\(ax^2 + bx + c\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = 4\)[/tex], and [tex]\(c = -12\)[/tex].
2. Find two numbers that multiply to [tex]\(a \cdot c\)[/tex] and add to [tex]\(b\)[/tex]:
- First, calculate [tex]\(a \cdot c = 1 \cdot (-12) = -12\)[/tex].
- We need two numbers whose product is [tex]\(-12\)[/tex] and whose sum is [tex]\(4\)[/tex]. These numbers are [tex]\(6\)[/tex] and [tex]\(-2\)[/tex] because [tex]\(6 \times (-2) = -12\)[/tex] and [tex]\(6 + (-2) = 4\)[/tex].
3. Rewrite the middle term using the two numbers:
- Rewrite [tex]\(4x\)[/tex] as [tex]\(6x - 2x\)[/tex]:
[tex]\(x^2 + 4x - 12 = x^2 + 6x - 2x - 12\)[/tex].
4. Group the terms for factoring by grouping:
- Group the terms in pairs:
[tex]\((x^2 + 6x) + (-2x - 12)\)[/tex].
5. Factor out the common factor from each pair:
- In the first group [tex]\((x^2 + 6x)\)[/tex], the common factor is [tex]\(x\)[/tex]:
[tex]\(x(x + 6)\)[/tex].
- In the second group [tex]\((-2x - 12)\)[/tex], the common factor is [tex]\(-2\)[/tex]:
[tex]\(-2(x + 6)\)[/tex].
6. Factor out the common binomial:
- Both groups now contain the common binomial factor [tex]\((x + 6)\)[/tex]:
[tex]\(x(x + 6) - 2(x + 6)\)[/tex].
7. Combine the factors:
- Factor out [tex]\((x + 6)\)[/tex] from both terms:
[tex]\((x - 2)(x + 6)\)[/tex].
Thus, the factorization of the quadratic expression [tex]\(x^2 + 4x - 12\)[/tex] is:
[tex]\[
(x - 2)(x + 6)
\][/tex]
