To match each quadratic function given in factored form with its equivalent standard form, we need to expand each factored form and compare it with the standard forms provided. Let's proceed step by step.
### 1. Expanding Factored Forms
a. [tex]\( (x + 2)(x - 6) \)[/tex]
[tex]\[
(x + 2)(x - 6) = x(x - 6) + 2(x - 6)
= x^2 - 6x + 2x - 12
= x^2 - 4x - 12
\][/tex]
b. [tex]\( (x - 4)(x + 3) \)[/tex]
[tex]\[
(x - 4)(x + 3) = x(x + 3) - 4(x + 3)
= x^2 + 3x - 4x - 12
= x^2 - x - 12
\][/tex]
c. [tex]\( (x - 12)(x + 1) \)[/tex]
[tex]\[
(x - 12)(x + 1) = x(x + 1) - 12(x + 1)
= x^2 + x - 12x - 12
= x^2 - 11x - 12
\][/tex]
d. [tex]\( (x - 3)(x + 4) \)[/tex]
[tex]\[
(x - 3)(x + 4) = x(x + 4) - 3(x + 4)
= x^2 + 4x - 3x - 12
= x^2 + x - 12
\][/tex]
### 2. Matching Standard Forms
Standard Forms:
A. [tex]\( x^2 - 11x - 12 \)[/tex]
B. [tex]\( x^2 - 4x - 12 \)[/tex]
C. [tex]\( x^2 + x - 12 \)[/tex]
D. [tex]\( x^2 - x - 12 \)[/tex]
Factored Forms and Expanded Results:
1. [tex]\( (x + 2)(x - 6) = x^2 - 4x - 12 \)[/tex]
2. [tex]\( (x - 4)(x + 3) = x^2 - x - 12 \)[/tex]
3. [tex]\( (x - 12)(x + 1) = x^2 - 11x - 12 \)[/tex]
4. [tex]\( (x - 3)(x + 4) = x^2 + x - 12 \)[/tex]
### 3. Final Matching:
[tex]\[
\begin{array}{llll}
A & \rightarrow & (x - 12)(x + 1) & : x^2 - 11x - 12 \\
B & \rightarrow & (x + 2)(x - 6) & : x^2 - 4x - 12 \\
C & \rightarrow & (x - 3)(x + 4) & : x^2 + x - 12 \\
D & \rightarrow & (x - 4)(x + 3) & : x^2 - x - 12 \\
\end{array}
\][/tex]
So, the matches are:
A. [tex]\( f(x) = x^2 - 11x - 12 \)[/tex] matches with [tex]\( (x - 12)(x + 1) \)[/tex]
B. [tex]\( f(x) = x^2 - 4x - 12 \)[/tex] matches with [tex]\( (x + 2)(x - 6) \)[/tex]
C. [tex]\( f(x) = x^2 + x - 12 \)[/tex] matches with [tex]\( (x - 3)(x + 4) \)[/tex]
D. [tex]\( f(x) = x^2 - x - 12 \)[/tex] matches with [tex]\( (x - 4)(x + 3) \)[/tex]
