Quadratic Functions and the Completing the Square Technique (Part 1)
(This page is Part 1. Click here for Part 2.)
You may want to review:
Let $\,a\,,$ $\,b\,$ and $\,c\,$ be real numbers, with $\,a \ne 0\,.$
Comments on the Definition
What type of term must a quadratic function have?
If $\,a = 0\,$ in the expression $\,ax^2 + bx + c\,,$ then the $\,x^2\,$ term disappears. Thus, the requirement that $\,a\ne 0\,$ says that a quadratic function must have an $\,x^2\,$ term.
What other types of terms is a quadratic function allowed to have?
There are no restrictions placed on $\,b\,$ or $\,c\,$ in the definition of quadratic function; they can be any real number.
In particular, $\,b\,$ is allowed to equal zero, which causes the $\,x\,$ term to disappear. Thus, a quadratic function may or may not have an $\,x\,$ term.
Similarly, $\,c\,$ is allowed to equal zero, which causes the constant term to disappear. Thus, a quadratic function may or may not have a constant term.
Summary of Terms Types in a Quadratic Function
A quadratic function must have an $\,x^2\,$ term; it is allowed to have an $\,x\,$ term; it is allowed to have a constant term. It may not have any other types of terms.
Quadratic Functions Go By Different Names
Like all expressions, $\,ax^2 + bx + c\,$ has lots of different names. Standard form and vertex form are just two names that are popular (since they're quite useful).
Different names are good for different purposes, so you should be able to switch from one to the other.
Going from vertex form, $\,a(x-h)^2 + k\,,$ to standard form is easy—just multiply out. Going from standard form, $\,ax^2 + bx + c\,,$ to vertex form is harder, and requires a technique that is explored in this section—the ‘completing the square technique’.
Why is the name ‘vertex form’ appropriate?
As derived in Equations of Simple Parabolas, equations of the form $\ y = a{(x-h)}^2 + k\ $ graph as parabolas, where $\,(h,k)\,$ is the vertex of the parabola. Hence, the name vertex form is appropriate, since it's easy to read the coordinates of the vertex from vertex form.
What does the graph of a quadratic function look like?
Every quadratic function graphs as a parabola with directrix parallel to the $x$-axis.
If $\,a \gt 0\,$ then the parabola is concave up (holds water).
If $\,a \lt 0\,$ then the parabola is concave down (sheds water).
Before discussing the technique of completing the square, a preliminary definition is needed:
All of the following are perfect square trinomials:
$$ \begin{gather} \cssId{s47}{x^2 + 2x + 1 = {(x+1)}^2}\cr\cr \cssId{s48}{x^2 - 6x + 9 = {(x-3)}^2}\cr\cr \cssId{s49}{x^2 + \frac{4}{5}x + \frac{4}{25} = {\bigl(x + \frac{2}{5}\bigr)}^2} \end{gather} $$shows that there is a very simple relationship between the coefficient of the $\,x\,$ term ($\,2k\,$) and the constant term ($\,k^2\,$) in a perfect square trinomial: take the coefficient of the $\,x\,$ term, divide it by $\,2\,,$ and then square the result, to obtain the constant term!
In other words, how can you get from $\,2k\,$ to $\,k^2\,$? Divide by two, and then square: $$ \cssId{s58}{2k} \ \ \cssId{s59}{\overset{\text{divide by } 2}{\rightarrow}} \ \ \cssId{s60}{k} \ \ \cssId{s61}{\overset{\text{square}}{\rightarrow}} \ \ \cssId{s62}{k^2} $$ This is the key observation in the following technique:
The process of finding the correct number to add to an expression of the form $\ x^2+bx\ $ to form a perfect square trinomial is called completing the square.
The correct number to add is: $$\cssId{s69}{ {\left(\frac{b}{2}\right)}^2}$$
That is, take the coefficient of the $\,x\,$ term, divide it by $\,2\,,$ and then square the result.
Then:
$$ \begin{align} &\cssId{s72}{\overset{\text{start with this }}{\overbrace{x^2 + bx}}} \cssId{s73}{+ \overset{\text{and add this number}}{\overbrace{ {\left(\frac{b}{2}\right)}^2}}}\cr\cr &\qquad\qquad \cssId{s74}{= \overset{\text{to get a perfect square}}{\overbrace{ {\left(x + \frac{b}{2}\right)}^2}}} \end{align} $$Example (Completing the Square)
Question: What number must be added to $\ x^2 - 3x\ $ to form a perfect square trinomial? Then, what perfect square trinomial results?
Solution: The $\,x\,$ term is $\,-3x\,.$ The coefficient of the $\,x\,$ term is $\,-3\,.$ Take this number, divide it by $\,2\,,$ and then square it. Thus, the number to add is $\ {(\frac{-3}{2})}^2\,.$
The perfect square trinomial that results is: $$\cssId{s86}{x^2 - 3x + \bigl(\color{red}{\frac{-3}{2}}\bigr)^2 = {\bigl(x\color{blue}{-\frac{3}{2}}\bigr)}^2}$$
Notice that the number inside the parentheses gets plopped down next to $\,x\,$ in the resulting perfect square:
$$ \cssId{s88}{x^2-3x} \cssId{s89}{+\bigl( \overset{\text{the # inside}}{\overbrace{ \frac{-3}{2}}} \bigr)^2} \cssId{s90}{\overset{\rightarrow \rightarrow \rightarrow \rightarrow}{ \vphantom{\left(\frac{ { { { { -3}^2}^2}^2}^2}{2}\right)} = } } \cssId{s91}{\bigl(\ \ x\overset{\text{goes here } }{\overset{\downarrow}{-\ \frac{3}{2 } } }\bigr)^2} $$