To factor the quadratic expression [tex]\( 3x^2 - 11x - 20 \)[/tex]:
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], and [tex]\( c = -20 \)[/tex].
2. Find two numbers that multiply to [tex]\( ac \)[/tex]:
- Here, [tex]\( a \cdot c = 3 \cdot (-20) = -60 \)[/tex].
- We need to find two numbers that multiply to [tex]\(-60\)[/tex] and add up to [tex]\( b = -11 \)[/tex].
3. Determine these two numbers:
- The numbers that fit these conditions are [tex]\( -15 \)[/tex] and [tex]\( 4 \)[/tex].
- This is because [tex]\(-15 \times 4 = -60\)[/tex] and [tex]\(-15 + 4 = -11\)[/tex].
4. Rewrite the middle term:
- Express the quadratic expression by splitting the middle term:
[tex]\[
3x^2 - 15x + 4x - 20
\][/tex]
5. Group the terms:
- Group the terms into two pairs:
[tex]\[
(3x^2 - 15x) + (4x - 20)
\][/tex]
6. Factor out the greatest common factor (GCF) from each pair:
- From the first pair [tex]\(3x^2 - 15x\)[/tex], factor out [tex]\(3x\)[/tex]:
[tex]\[
3x(x - 5)
\][/tex]
- From the second pair [tex]\(4x - 20\)[/tex], factor out [tex]\(4\)[/tex]:
[tex]\[
4(x - 5)
\][/tex]
7. Combine the factored expressions:
- Now the expression looks like:
[tex]\[
3x(x - 5) + 4(x - 5)
\][/tex]
- Notice that [tex]\((x - 5)\)[/tex] is a common factor.
8. Factor out the common binomial [tex]\((x - 5)\)[/tex]:
- Combine the terms:
[tex]\[
(x - 5)(3x + 4)
\][/tex]
Thus, the fully factored form of the quadratic expression [tex]\(3x^2 - 11x - 20\)[/tex] is:
[tex]\[
(x - 5)(3x + 4)
\][/tex]
