Completing the Square

Tags

math

Is a special reformulation of a quadratic equation into the form \[ a(x+d)^2 + e \] This form has some special properties that makes it preferable to the expanded form in some situations.

Properties include:

1. \( (-d, e) \) is the minimum (or maximum) point of the line formed by \( y \).
2. \( \begin{pmatrix}-ad \\ e\end{pmatrix} \) is the vector of translation from the line \( y=x^2 \).
3. The line of symmetry of \( y \) is \( x = -d \).
4. The real roots of the equation \( y \) are easily visible \( x = -d \pm \sqrt{\frac{-e}{a}} \).

Formulation

We can complete the square by:

1. Factorising out any common factors of the \( ax^2 + bx \) term.
2. Factorising the body of step 1, into \( (x+n)^2 - m \) where \( m = e - n^2 \).
3. Simplify the result.

For example:

\begin{align} 3x^2 -6x + 5 &= 3[x^2 - 2x] + 5 \\

           &= 3 [(x - 1)^2 - 1] + 5 \\\\
           &= 3(x-1)^2 - 3 + 5 \\\\
           &= 3(sx-1)^2 +2 \label{eq:complete-square-eg}

\end{align}

The minimum point of eq:complete-square-eg is \( (1, 2) \).