shankar.ashwin wrote:If the roots of the quadratic equation ax^2 + bx +2 = 0 are integers, how many possible values could (a,b) take given that 'a' and 'b' are integers and a>2 and b<6 ?
A) None
B) 1
C) 2
D) 8
E) More than 8
Given ax²+bx+c=0, the product of the roots = c/a and the sum of the roots = -b/a.
Since the roots are integers, their product (c/a) and their sum (-b/a) must also be integers.
If ax²+bx+2=0, then c=2, implying that the product of the roots = 2/a.
Since a≥2, and 2/a is an integer, we know that a=2 and that the product of the roots = c/a = 2/2 = 1.
Since the roots are integers, there are only two possible cases:
Both roots are 1 (so that their product is 1*1 = 1 and their sum is 1+1 = 2).
Both roots are -1 (so that their product is -1 * -1 = 1 and their sum is -1 + -1 = -2).
If the sum of the roots is 2, then -b/2 = 2 and b=-4.
If the sum of the roots is -2, then -b/2 = -2 and b=4.
Thus, there are two possibilities for (a,b):
(2,4) and (2,-4).
The correct answer is
C.
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