Sure, let's factor the given expression completely. Let's consider the expression [tex]\( x^2 - 8x - 48 \)[/tex].
Step-by-step solution:
1. Identify the quadratic expression: The quadratic expression we need to factor is [tex]\( x^2 - 8x - 48 \)[/tex].
2. Look for two numbers that multiply to the constant term (-48) and add to the coefficient of the linear term (-8): We need to find two numbers that multiply to [tex]\(-48\)[/tex] and add up to [tex]\(-8\)[/tex].
- Factors of [tex]\(-48\)[/tex] include: [tex]\((-1, 48), (1, -48), (-2, 24), (2, -24), (-3, 16), (3, -16), (-4, 12), (4, -12), (-6, 8), (6, -8)\)[/tex].
- Out of these, the pair that adds up to [tex]\(-8\)[/tex] is [tex]\((-12)\)[/tex] and [tex]\(4\)[/tex], because [tex]\((-12) + 4 = -8\)[/tex].
3. Rewrite the middle term using these numbers: Modify the quadratic expression to include these two numbers:
[tex]\[
x^2 - 8x - 48 = x^2 - 12x + 4x - 48.
\][/tex]
4. Group the terms: Group the expression into two pairs of terms:
[tex]\[
x^2 - 12x + 4x - 48 = (x^2 - 12x) + (4x - 48).
\][/tex]
5. Factor out the greatest common factor (GCF) from each group:
[tex]\[
= x(x - 12) + 4(x - 12).
\][/tex]
6. Factor out the common binomial factor: Notice that [tex]\((x - 12)\)[/tex] is a common factor.
[tex]\[
= (x - 12)(x + 4).
\][/tex]
So, the completely factored form of the expression [tex]\( x^2 - 8x - 48 \)[/tex] is:
[tex]\[
(x - 12)(x + 4).
\][/tex]
Therefore, the answer is [tex]\((x - 12)(x + 4)\)[/tex].
