Sure, let's factorize the quadratic expression step-by-step.
Given the quadratic expression: [tex]\( 2x^2 - 11x - 21 \)[/tex]
To factorize the quadratic expression, we look for two binomials of the form:
[tex]\[ (Ax + B)(Cx + D) \][/tex]
Multiplying these binomials, we get:
[tex]\[ (Ax + B)(Cx + D) = ACx^2 + (AD + BC)x + BD \][/tex]
We need this to match the form [tex]\( 2x^2 - 11x - 21 \)[/tex]. Therefore:
1. [tex]\( AC = 2 \)[/tex]
2. [tex]\( AD + BC = -11 \)[/tex]
3. [tex]\( BD = -21 \)[/tex]
Now, we choose [tex]\( A \)[/tex], [tex]\( B \)[/tex], [tex]\( C \)[/tex], and [tex]\( D \)[/tex] such that these conditions are satisfied.
Let's find the pairs of factors for [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex]:
- Since [tex]\( AC = 2 \)[/tex], the possible pairs for [tex]\((A, C)\)[/tex] are [tex]\((1, 2)\)[/tex] or [tex]\((2, 1)\)[/tex].
- Since [tex]\( BD = -21 \)[/tex], the possible pairs for [tex]\((B, D)\)[/tex] are [tex]\((1, -21), (-1, 21), (3, -7), (-3, 7), (7, -3), (-7, 3)\)[/tex].
We need to select the correct pairs so that [tex]\( AD + BC = -11 \)[/tex].
After examining possible pairs, we find:
- When [tex]\( A = 1 \)[/tex], [tex]\( C = 2 \)[/tex], [tex]\( B = -7 \)[/tex], [tex]\( D = 3 \)[/tex]:
- [tex]\( AD + BC = (1)(3) + (-7)(2) = 3 - 14 = -11 \)[/tex]
So, the factorization is given by:
[tex]\[ 2x^2 - 11x - 21 = (x - 7)(2x + 3) \][/tex]
Therefore, the complete factorization is:
[tex]\[ (x - 7)(2x + 3) \][/tex]
So, the blank terms should be:
[tex]\[ (\quad x - 7 \quad ) ( \quad 2x + 3 \quad ) \][/tex]
