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.
+119 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?
+129899 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
+119 Yep, this certainly works as well. Kinda just reversing my steps and then finishing the problem. :)
