To complete the square for the polynomial \(f^2 + 28f\), we need to find a number that makes it a perfect-square quadratic.
-
Take the coefficient of the linear term \(28\), divide it by 2, and square it: \[ \left(\frac{28}{2}\right)^2 = 14^2 = 196 \]
-
Now, add this number \(196\) to the polynomial: \[ f^2 + 28f + 196 \]
Thus, the completed square is: \[ f^2 + 28f + 196 = (f + 14)^2 \]
So, the number you need to fill in to make the polynomial a perfect-square quadratic is 196.