Question

match each quadratic function given in factored form with its equivalent standard form listed.

on the left (inside a gray rectangle) are four quadratic functions in standard form:

• ( f(x) = x^2 - 4x - 12 )
• ( f(x) = x^2 - 11x - 12 )
• ( f(x) = x^2 + x - 12 )
• ( f(x) = x^2 - x - 12 )

on the right, four quadratic functions in factored form are listed vertically, each with a dashed rectangle above it and a line connecting to a blank dashed rectangle (presumably for matching):

• ( f(x) = (x - 12)(x + 1) )
• ( f(x) = (x - 3)(x + 4) )
• ( f(x) = (x - 4)(x + 3) )
• ( f(x) = (x + 2)(x - 6) )

Explanation:

Step1: Expand factored form 1

$$\begin{align*} (x-12)(x+1)&=x^2+x-12x-12\\ &=x^2-11x-12 \end{align*}$$

Step2: Expand factored form 2

$$\begin{align*} (x-3)(x+4)&=x^2+4x-3x-12\\ &=x^2+x-12 \end{align*}$$

Step3: Expand factored form 3

$$\begin{align*} (x-4)(x+3)&=x^2+3x-4x-12\\ &=x^2-x-12 \end{align*}$$

Step4: Expand factored form 4

$$\begin{align*} (x+2)(x-6)&=x^2-6x+2x-12\\ &=x^2-4x-12 \end{align*}$$

Step5: Match expanded and standard forms

Pair each result with the corresponding standard quadratic.

Answer:

1. $f(x)=(x-12)(x+1) \iff f(x)=x^2-11x-12$
2. $f(x)=(x-3)(x+4) \iff f(x)=x^2+x-12$
3. $f(x)=(x-4)(x+3) \iff f(x)=x^2-x-12$
4. $f(x)=(x+2)(x-6) \iff f(x)=x^2-4x-12$