If k is a positive integer less than 17, what is the total number of possible integer solutions for the equation `x^(2)+8x+k=0`?

Options

A. `5`
B. `6`
C. `7`
D. `8`

Answer

C

Explanation

Difficult: Hard
Category: Passport to Advanced Math/ Quadratics
Strategy Advice: When a quadratic equation is factored, the value of the constant (here, k) is equal to the product of the two constants in the binomial factors. Determine how many combinations there are, and you'll have your answer.
Getting to the Answer: Imagine factoring the given equation: You would need to find the factors of k (which you're told is less than 17) that add up to 8. List all possible integer combination whose sum is 8 and whose product is less than 17:
`(x+1)(x+7)=0:2` solutions
`(x+2)(x+6)=0:2` solutions
`(x+3)(x+5)=0:2` solutions
`(x+4)(x+4)=0:2` solutions
There are 7 different integer solutions for this equation, which matches (C).