Alani gives you an Algebra puzzle as follows:

Question

The expression d2 − d − 6 can be written in factored form as (d + 2)(d + k), where k represents a number. What is the value of k?

Group of answer choices

3

-2

-3

2

Answer 1

To factor the expression \( d^2 - d - 6 \) into the form \( (d + 2)(d + k) \), we need to find a value for \( k \).

1. First, let's expand the expression \( (d + 2)(d + k) \): \[ (d + 2)(d + k) = d^2 + kd + 2d + 2k = d^2 + (k + 2)d + 2k \]

2. Now we want this expression to equal \( d^2 - d - 6 \): \[ d^2 + (k + 2)d + 2k = d^2 - d - 6 \]

3. By comparing coefficients, we can set up the following equations:

   • Coefficient of \( d \): \( k + 2 = -1 \)
   • Constant term: \( 2k = -6 \)

4. Let's solve the first equation for \( k \): \[ k + 2 = -1 \implies k = -1 - 2 = -3 \]

5. Now, let's check the second equation with \( k = -3 \): \[ 2k = -6 \implies 2(-3) = -6 \implies -6 = -6 \] This is true.

Thus, the value of \( k \) is: \[ \boxed{-3} \]