What's the total number of possible integer solutions for this equation?

Question

If k is a positive integer less than 17, what's the total number of possible integer solutions for the equation x²+8x+k=0?

Answer 1

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]
 
  “How many integers k with values 1-16 satisfy x² + 8x + k = 0 ?”

Now, remember F-O-I-L:
      (x+a)(x+b) = x² + (a+b)x + ab

So, 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

   5   3      15 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. 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.