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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.

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 \( n = -3\).

Note by Arron Kau
3 years, 2 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. Anurag Pandey · 9 months, 4 weeks ago

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