Determine the values of k so that `9w^2-kw+36` is factorable:
There are multiple answers to this question depending on the set of numbers you are factoring over:
(1) Complex numbers: this will always factor. The fundamental theorem of algebra guarantees that every polynomial factors into linear factors.
(2) Real numbers: the analysis by jeew-m is correct. Any value of k where the determinant is greater than or equal to zero will allow factoring over the reals.
(3) Rationals: Let p and q be numbers so that their product is 324 . Then any k=p+q will be factorable over the rationals.
For example, let p=27 and q=12. Then k=39.
Here the roots are 3 and `4/3`
The possibilities are 325,164,111,85,60,45,39,36
(4) Integers: Here the discriminant must not only be positive, but also a perfect square number. k=45 or k=36 are the possibilities.