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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 \(a\) ?

 Oct 22, 2019
edited by Guest  Oct 22, 2019
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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

 

.

 Oct 22, 2019

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