x^2 + x -k?

Question

Find all positive values for k for which each of the following can be factored.
I need help getting to this answer. I am not sure which formula should I use to solve it. I am not looking for just an answer but more the proper way to solve it

Answer 1

the sum is -1
the product is -k

(x+1)(x-k)=x^2+(1-k)x-k

1-k=1
k=0


(x-1)(x+k)=x^2+(k-1)x-k
k-1=1
k=2

answer k=0 or k=2

Answer 2

Are you looking for real roots or rational roots?

For real roots (b^2 - 4ac)>= 0
or in this case
1+4k >=0

For rational roots, consider the expansion of
(x+a)(x+b)
= x^2 + ax + bx + ab
= x^2 + (a+b)x + ab
Compare this with the equation you have
x^2 terms match each other
a+b = 1
b = 1-a

ab = -k
a(1-a) = -k
a(a-1) = k
k must be the product of two rational numbers which differ by 1.
For roots that are integers, k must be the product of successive integers such as 6=2*3 or 12=3*4.

Answer 3

k is a positive number of the form n(n+1).
The expression could then be factored as
(x-n)(x+(n+1))
For example 1 x 2 = 2
2 x 3=6
3 x 4 =12
etc. would work.

Answer 4

x^2+x-k=0
discriminant=1+4k
for real roots:
1+4k>0
k>-1/4& then the given expression can be factored

Answer 5

x(x-k)