For the surd
The task is: What's a general statement which represents values for k which make the exact value an integer.
What does the question mean by 'general statement'? what form is my answer supposed to be in?
Now, if 1+4k is a perfect square, it will be an odd integer. And the square root of an odd integer is an odd integer. Added to 1, it'll give an even integer, which, divided by 2, will be an integer.
Is it possible to create some kind of formula like the used,
which only works for values of k that make x an integer?
The exact question is: Find the general statement that represnts all the values of
for which the expression is an integer.
So is all you need to completely satisfy that goal
simply state the conditions for , that is a perfect square? . . . . yes
We have: . , for some integer
. . Then: .
We see that:
Then  becomes: .
Hence, must be the product of two consecutive integers.
Then: . , an integer.
I hope there are other parts to this investigation which will allow at least some of the portfolio to reflect your own work.
Yes, I came to the conclusion through my own working that k must be the product of two consecutive numbers. I was just wondering whether when it asks for a 'general statement' wether it wants a written description for the conditions of k that make x an integer, or some formula like this one: which only produces integer values for x from certain values of k. I was thinking of substituting in one of these conditions for k, which is
(where a and b are consecutive)