(3 + root(a))(4 + root(a)) = 17 + k(root(a)) where a and k are positive integers. Find the value of a and the value of k.

Let's open out the bracket using the FOIL method (first, outside, inside, last):

(3 + root(a))(4 + root(a)) = 12 + 3root(a) + 4root(a) + (root(a))² = 12 + 7root(a) + a.

Since the answer 17 + k(root(a)) is in the form of an integer + surd, we must equate the integers and surds of 12 + 7root(a) + a     with       17 + k(root(a)).

Therefore, 12 + a = 17       so     a = 5

7root(a) = k(root(a))           so     k = 7.