Math
posted by Raj on .
Find the value of K such that the following trinomials can be factored over the integers:
1. 36x^2+18x+K
2. 3x^2  16x+K

let's look at the discriminant.
if b^2  4ac is a perfect square, then it can be factored over the rationals, so we start with that.
I will do the 2nd, since it has smaller numbers
for 3x^2  16x + k
b^2  4ac
= 25612k
= 4(643k)
remember we have to take the square root of that
√(4(643k))
= 2√(643k)
For 643k to be a perfect square
we need 3k to be 0,15,28,39,48,55,60 or 63
of those only 0,15,39,48,60, and 63 are multiples of 3
So for rationals, k could be 0, 5, 13, 16, 20 or 21
testing:
let's try k = 13
3x^2  16x + 13
x = (16 ± √100)/6
= 1 or 13/3
so 3x^2  16x + 13 = (x+1)(3x13)
k = 0, 5, 13, 16, 20, or 21