### 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.

### Worked Examples

## 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 \( n = -3\).

## 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. – Anurag Pandey · 11 months, 3 weeks ago

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