For how many integer values of $a$ does the equation $x^2 + ax + 5a = 0$ have integer solutions for $x$?
+128408 x^2 + ax + 5a = 0
Let m, n be the integer roots...this implies that....
(x - m) (x - n) = 0
x^2 - (m + n)x + mn = 0
So....equating terms.....we have this
-m - n = a ⇒ -5m -5n = 5a
mn = 5a
So this implies that
-5m - 5n = mn
-5n - mn = 5m
-n ( 5 + m) = 5m
n = -5m / ( m + 5)
Note that, as m → ± inf, n → -5
Also.....m must be a multiple of 5 since -5m is a multiple of 5
So we can test some values
m n -m - n = a
0 0 0
20 -4 -16
All positive values of m > 20 means that -5 < n < -4 so no other integer values are possible for in this direction
-10 -10 20
-30 -6 36
All negative values of m < -30 means that -6 < n < -5 so no other integer values are possible for n at all
So the possible values for "a" are :
-16, 0, 20 and 36
