Certainly! To factor the quadratic expression [tex]\(a^2 - 3a - 28\)[/tex], let's go through the steps systematically:
1. Identify the quadratic expression:
[tex]\[
a^2 - 3a - 28
\][/tex]
2. Look for factors of the constant term: In this case, the constant term is [tex]\(-28\)[/tex]. We need to find two numbers that multiply to [tex]\(-28\)[/tex].
3. Find pair of factors that add up to the middle coefficient: We need those factors to also sum to the middle coefficient, which is [tex]\(-3\)[/tex].
Let's list the pairs of factors of [tex]\(-28\)[/tex]:
[tex]\[
(1, -28), (-1, 28), (2, -14), (-2, 14), (4, -7), (-4, 7)
\][/tex]
4. Identify the correct pair of factors: Among these, the pair [tex]\((4, -7)\)[/tex] works because:
[tex]\[
4 \cdot (-7) = -28 \quad \text{and} \quad 4 + (-7) = -3
\][/tex]
5. Rewrite the quadratic expression: We can use these factors to split the middle term [tex]\(-3a\)[/tex] into two terms. Thus, we rewrite:
[tex]\[
a^2 - 3a - 28 = a^2 + 4a - 7a - 28
\][/tex]
6. Group terms to factor by grouping:
[tex]\[
a^2 + 4a - 7a - 28 = (a^2 + 4a) + (-7a - 28)
\][/tex]
7. Factor out the common factors in each group:
[tex]\[
a(a + 4) - 7(a + 4)
\][/tex]
8. Factor out the common binomial factor:
[tex]\[
(a + 4)(a - 7)
\][/tex]
So, the factored form of the quadratic expression [tex]\(a^2 - 3a - 28\)[/tex] is:
[tex]\[
(a - 7)(a + 4)
\][/tex]
