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 \(a\) ?
starting with the original expression x2 + ax = –14
add 14 to both sides to get a quadratic equation x2 + 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 x2 + 15x +14 ... in this instance a = 15
(x + 7)(x +2) gives us x2 + 9x + 14 ... in this instance a = 9
15 is larger than 9 so it looks like the answer is 15
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