# Completing the Square

### Definition

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 $n = -3$.

Note by Arron Kau
6 years, 5 months ago

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Could 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.

- 4 years, 1 month ago