For a quadratic polynomial \(f(x) = ax^2 + bx +c\), completing the square means giving an expression of the form
\[f(x) = A(x-B)^2 + C\]
\[-x^2 + 4x + 10 = - (x-2)^2 + 14\]
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 \( -x^2 + 4x + 10\)?
- \( -x^2 + 4x + 10 = - (x-2)^2 + 14\).
Since squares are non-negative,
- \( - (x-2)^2 + 14 \leq 14\).
Thus, the maximum value of the quadratic is \( 14\), which is achieved at \( x=2\).
2. For what integer value \( n\) is \( n^2 + 6x + 10\) also a perfect square?
Completing the square, we see that
- \(n^2 + 6x + 10 = (n+3)^2 +1\).
- \( n^2 + 6x + 10 = m^2\)
for some integer \( m\), then
- \( 1 = m^2 - (n+3)^2\).
The only perfect squares that differ by \( 1\) are \( 0\) and \(1\). Hence,
- \( (n+3)^2 = 0 \),
which has the solution \( n = -3\).