Express f(x) = x^2 + 5x + 9 in the form (x + a)^2 + b, stating the values of a and b.

The question asks you to complete the square on the function f(x). First, we split the function:

f(x) = (x^2 + 5x) + 9

And recognise that we need the form (x^2 + 2k + k^2). Identifying 2k = 5, we have that k = 5/2:

f(x) = (x^2 + 2*(5/2)x + (5/2)^2) + 9 - (5/2)^2

with the last term being due to the addition of (5/2)^2: completing the square. Resolving the expression:

f(x) = (x + 5/2)^2 + 11/4, with a = 5/2 and b = 11/4. Expanding the brackets again reforms the original expression.