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Find all integers n such that the quadratic 7x^2 + nx + 18 can be expressed as the product of two linear factors with integer coefficients.

 Nov 26, 2020
 #1
avatar+110 
+1

In order for this to work, you are going to want a setup of \((7x+a)(x+b)\), since that's the only way to attain the coefficient of the quadratic term. Expanding this gives \(7x^2+(a+7b)x+ab\). We have \(ab=18\) from setting coefficients equal and we also have an integer coefficients condition. Therefore, testing integer values of \(a,b\) will suffice. Can you take it from here?

 Nov 26, 2020
 #2
avatar+116126 
+1

All    factors  of  7  =   1 , 7

 

All   factors of 18   =  1, 18     2, 9    3, 6

 

So we have

 

(x + 1) ( 7x + 18)      n  = 25

( x + 18) ( 7x + 1)   n  =  127

 

( x + 2) (7x + 9)       n = 23

( x + 9) ( 7x  + 2)    n  =  65

 

( x + 3) (7x + 6)    n   = 27

( x + 6) ( 7x + 3)   n  =  45

 

 

cool cool cool

 Nov 26, 2020
 #3
avatar+110 
+1

Yep, this certainly works as well. Kinda just reversing my steps and then finishing the problem. :)

OlympusHero  Nov 27, 2020

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