Don´t really think that if the roots are prime numbers this would be possible, since numbers with decimals can´t count as prime. Negative numbers can´t be prime numbers (since they have another factor which is -1 and its number)
I thought of having roots 7 and 1. 7 is a prime number but 1 isn´t.
Now I´m wondering...
X1 . X2 = c/a = 7
Meaning that the two roots when multiplying must be 7, which means 2 prime numbers multiplying must be 7, which in my opinion is not possible.
The only option I have closest to the given conditions are 7 (prime number) and 1 (not a prime number).
In that case:
X1 = 7
x²-3KX+7=0
49 - 21K +7 = 0
21K = 56
K = 56 / 21 = 8 / 3
X2 = 1
x²-3KX+7=0
1 - 3K +7 = 0
3k = 8
K = 8/3
So the value of K is 8/3.
Note that this is the closest answer I can give to your given conditions since 1 is not considered as Prime number.
If both roots were really prime numbers, then I don´t think it´s possible to solve K.