Often, the key to solving a problem is to simplify, subdivide, and rephrase it (i.e., “put it into your own words”).With this problem, note:

“k is a positive integer” [0 is not positive]

“less than 17” [limits the options to integers 1-16]

Now, remember F-O-I-L:

(x+a)(x+b) = x

So, putting this problem in my own words,

Note that enumerating (listing) and counting them just got a lot easier.

a b 1<=(ab)<=16 ?

1 7 7 yes

“k is a positive integer” [0 is not positive]

“less than 17” [limits the options to integers 1-16]

**“How many integers k with values 1-16 satisfy x**^{2}+ 8x + k = 0 ?”Now, remember F-O-I-L:

(x+a)(x+b) = x

^{2}+ (a+b)x + abSo, putting this problem in my own words,

**“How many sets of integers (a,b) are there that add to 8 and multiply to an integer 1-16?”**[important note: A “solution” is a value of x that makes the equation zero; “possible values of k” is a better way to ask this question.]Note that enumerating (listing) and counting them just got a lot easier.

a b 1<=(ab)<=16 ?

1 7 7 yes

2 6 12 yes

3 5 15 yes

4 4 16 yes

3 5 15 yes

4 4 16 yes

5 3 15 yes

6 2 12 yes

6 2 12 yes

7 1 7 yes

Reviewing the problem to ensure that we answered it: “how many possible values of k” are there? They are: 7, 12, 15, 16.

However, if the problem really intended to use the word “solve” meaning “values of x that makes the equation true for an integer value of k of 1-16”.” Then the “solutions” are 1, 2, 3, 4, 5, 6, 7. That’s 7 (of course) and this interpretation seems super simple.

Reviewing the problem to ensure that we answered it: “how many possible values of k” are there? They are: 7, 12, 15, 16.

**That’s 4.**However, if the problem really intended to use the word “solve” meaning “values of x that makes the equation true for an integer value of k of 1-16”.” Then the “solutions” are 1, 2, 3, 4, 5, 6, 7. That’s 7 (of course) and this interpretation seems super simple.

Kenneth S.

11/25/16