For a quadratic polynomial , completing the square means giving an expression of the form
We end up with a square term and a constant, both of which are easier for us to understand. This gives us an easy way to understand the graphs of quadratic polynomials, and to graph parabolas.
1. What is the maximum value of ?
Since squares are non-negative,
Thus, the maximum value of the quadratic is , which is achieved at .
2. For what integer value is also a perfect square?
Completing the square, we see that
for some integer , then
The only perfect squares that differ by are and . Hence,
which has the solution .