The equation \(x^2+ ax = -14\) has only integer solutions for \(x\) . If \(a\) is a positive integer, what is the greatest possible value of \(a\) ?

Guest Oct 22, 2019

edited by
Guest
Oct 22, 2019

#1**0 **

starting with the original expression x^{2} + ax = –14

add 14 to both sides to get a quadratic equation x^{2} + ax + 14 = 0

we're going to factor that quadratic equation

observe that there are only two ways to get 14 ... namely 7*2 or 14*1

we could use –7 times –2 or –14 times –1 but that would cause a to be negative so we discard that idea

try them both (x + 14)(x + 1) gives us x^{2} + 15x +14 ... in this instance a = 15

(x + 7)(x +2) gives us x^{2} + 9x + 14 ... in this instance a = 9

15 is larger than 9 so it looks like the answer is 15

.

Guest Oct 22, 2019