To factor the quadratic expression [tex]\(3x^2 - 11x - 20\)[/tex], follow these steps:
1. Identify the coefficients:
The given quadratic expression is in the form [tex]\(ax^2 + bx + c\)[/tex], where:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = -11\)[/tex]
- [tex]\(c = -20\)[/tex]
2. Look for factors:
We need to find two numbers that multiply to [tex]\(a \cdot c = 3 \cdot (-20) = -60\)[/tex] and add up to [tex]\(b = -11\)[/tex].
3. Find factor pairs:
The pairs of factors of -60 that add up to -11 are:
[tex]\((-15) \cdot 4 = -60\)[/tex] and [tex]\((-15) + 4 = -11\)[/tex].
4. Rewrite the middle term:
Rewrite [tex]\(-11x\)[/tex] using these two numbers:
[tex]\[
3x^2 - 15x + 4x - 20
\][/tex]
5. Group terms:
Group the terms to factor by grouping:
[tex]\[
(3x^2 - 15x) + (4x - 20)
\][/tex]
6. Factor out the greatest common factor (GCF) from each group:
[tex]\[
3x(x - 5) + 4(x - 5)
\][/tex]
7. Factor by grouping:
Notice that [tex]\((x - 5)\)[/tex] is a common factor:
[tex]\[
(3x + 4)(x - 5)
\][/tex]
We've factored the quadratic expression [tex]\(3x^2 - 11x - 20\)[/tex] into [tex]\((3x + 4)(x - 5)\)[/tex].
Thus, the factored form of the expression is:
[tex]\[
(3x + 4)(x - 5)
\][/tex]
