SOLUTION: If two prime numbers are roots of the equation x^2-12x+k=0, what is the value of k?

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Question 1074883: If two prime numbers are roots of the equation x^2-12x+k=0, what is the value of k?
Found 2 solutions by Boreal, ikleyn:
Answer by Boreal(15235)   (Show Source): You can put this solution on YOUR website!
the discriminant is sqrt (144-4k), and this has to be a perfect square.
the root is 12+/- sqrt (144-4k) all divided by 2
prime numbers produced by 6+/- x occur when x is odd.
they are limited to 5 and 7 as the roots, when the discriminant is 4, the sqrt of it= 2, and divided by 2 is 1. For the discriminant to be 4, b^2-4ac has to be 4, and that occurs when k is 35.
other roots would be 4,8, neither prime
3 and 9, 2 and 10 don't work.
x^2-12x+35=0
Answer by ikleyn(52785)   (Show Source): You can put this solution on YOUR website!
.
The symmetry axis for this parabola is x = 6, for any value of k.


Therefore, for any value of k, the roots are equidistant from the point x = 6.


Since we are looking for integer positive roots, we can have only these pairs:

(5,7)

(4,8)

(3,9)

(2,10)

(1,11)


Of them, only the pair (5,7) satisfies the given condition.


Answer.  Only the pair (5,7) satisfies the given condition, which gives k = 5*7 = 35.


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