Given the points (2,k) and (0,−6) for which values of k would the distance between the points be square root of 5
?

So, the distance between two points (x1,y1) and (x2,y2) is:

d = sqrt( (x2-x1)^2 + (y2-y)^2 )

Some times is better the square of the distance:

d^2 = (x2-x1)^2 + (y2-y)^2

Now, in your case:

5^2 = (0-2)^2 + ( -6-k)^2 = (-2)^2 + (6+k)^2 = 4 + (6+k)^2

Notice the little thing about (-6-k)^2 = (6+k)^2 (less '-') Finally:

25 = 4 + (6+k)^2 ===> (6+k)^2 = 21,

6+k  = sqrt(21)

Notice we did not expand (6+k)^2! No need this time. But now, the tricky part!

6+k=+sqrt(21) and 6+k=-sqrt(21), because both work (never forget that sqrt come with +/- when looking for solutions!)

k = -6 + sqrt(21) and k=-6-sqrt(21).

Both are valid answers. Some times one is not good because you need the solution be positive (for instance), but here, both are good.