For a quadratic polynomial \(f(x) = ax^2 + bx +c\), completing the square means giving an expression of the form

$f(x) = A(x-B)^2 + C$

For example,

$-x^2 + 4x + 10 = - (x-2)^2 + 14$

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 $-x^2 + 4x + 10$?

From above,

- $-x^2 + 4x + 10 = - (x-2)^2 + 14$.
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

- $n^2 + 6x + 10 = (n+3)^2 +1$.
If

- $n^2 + 6x + 10 = m^2$
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 $n = -3$.

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## Comments

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TopNewestCould you explain how to make graph of y=f( k - x ) where y= $ax^2 + bx +c$ Or in that using the curve y=$x^2 - 9x + 20$. ? Where k is some constant.

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