f(x) = x^2 + 2x + c?

Question

Find the value of the constant c for which the range given by f(x) >=3.

Answer 1

First, complete the square on f(x):

f(x) = x² + 2x + 1 + c - 1
f(x) = (x+1)² + (c-1)

Now, it is obvious that the range of this function is {x: x≥c-1}, so we just want the value such that c-1 = 3, which is c=4. And we are done.