Find all positive values for k for which the following
can be factored.
x²+ x - k
It can be factored if the discriminant B²-4AC is the
square of an integer. A=1, B=1, C=-k
B²-4AC = 1²-4(1)(-k) = 1+4k
We see that 1+4k is an odd number, and can be any odd
perfect square. But any odd perfect square is the
square of an odd integer.
Any positive odd number can be represented by 2n+1
where n is a non-negative integer, so
1 + 4k = (2n+1)²
1 + 4k = 4n² + 4n + 1
4k = 4n² + 4n
k = n² + n
k = n(n+1) for any non-negative integer.
But since k can't be 0, we have to rule out n=0
and change it to
k = n(n+1) for any POSITIVE integer n.
So k must be any term of this sequence 2, 6, 12, 20, 30, ...
Edwin
