1.1.1 Completing the Square

In this topic we will learn how to:

• carry out the process of completing the square for a quadratic polynomial $ax^{2} + bx + c$ and use a completed square form

There are two main scenarios, when completing the square. The first scenario is when the coefficient of $x^{2}$ is $1$ and the second scenario is when the coefficient of $x^{2}$ is not $1$. Let’s see how we go about the two scenarios:

$$\textcolor{gray}{\textbf{Completing the square when the}\\ \textbf{ coefficient of }x^{2}\textbf{ is }1}$$The general form for completing the square when the coefficient of $x^{2}$ is $1$, where the quadratic is $x^{2} + bx + c$ :
$$\left(x + \frac{b}{2}\right)^{2} - \left(\frac{b}{2}\right)^{2} + c$$Let’s look at an example.

Example 1
Complete the square for $x^{2} + 5x - 6$.

The first step is to rearrange the equation into the form $x^{2} + bx + c$,
$$x^{2} + \textcolor{#2192ff}{5}x \textcolor{#0f0}{-\ 6}$$Identify $b$ and $c$,
$$\textcolor{#2192ff}{b} = \textcolor{#2192ff}{5}, \ \ \textcolor{#0f0}{c} = \textcolor{#0f0}{-6}$$
Divide $\textcolor{#2192ff}{b}$ by $2$,
$$\textcolor{#2192ff}{\frac{5}{2}}$$Substitute into the general completed square form,
$$\left(x + \textcolor{#2192ff}{\frac{5}{2}}\right)^{2} - \left(\textcolor{#2192ff}{\frac{5}{2}}\right)^{2} + \textcolor{#0f0}{(- 6)}$$Simplify,
$$\left(x + \frac{5}{2}\right)^{2} -\frac{49}{4}$$

$$\textcolor{gray}{\textbf{Completing the square when the}\\ \textbf{ coefficient of }x^{2}\textbf{ is not 1}}$$Let’s look at an example.

Example 2
Complete the square for $2x^{2} + 9x + 10$.

The first step is to rearrange the equation into the form $ax^{2} + bx + c$,
$$2x^{2} + 9x + 10$$Factor out the coefficient of $x^{2}$ from the first two terms,
$$2\textcolor{#2192ff}{\left(x^{2} + \frac{9}{2}x\right)} + 10$$Complete the square for the quadratic inside parentheses using the method outlined above i.e Completing the square when the coefficient of $x^{2}$ is $1$,
$$2\textcolor{#2192ff}{\left[\left(x + \frac{9}{4}\right)^{2}- \left(\frac{9}{4}\right)^{2}\right]} + 10$$Simplify inside the square brackets,
$$2\textcolor{#2192ff}{\left[\left(x + \frac{9}{4}\right)^{2}- \frac{81}{16}\right]} + 10$$Remove the square brackets,
$$2\left(x + \frac{9}{4}\right)^{2} - \frac{81}{8} + 10$$Simplify,
$$2\left(x + \frac{9}{4}\right)^{2} - \frac{1}{8}$$

$$\textcolor{gray}{\textbf{Locating the vertex from the}\\ \textbf{ completed square form}}$$We can also use the completed square form to determine the coordinates of the vertex.

Given the completed square form, $(x + b)^{2} + c$, the vertex is,
$$(-b, c)$$Note: The vertex is also known as the turning point, stationary point, maximum or minimum point.

Let’s look at some past paper questions on completing the square

1. Functions $f$ and $g$ are both defined for $x \in \mathbb{R}$ and are given by
$$f(x) = x^{2} - 4x + 9$$$$g(x) = 2x^{2} + 4x + 12$$Express $f(x)$ in the form $(x - a)^{2} + b$. (9709/11/O/N/22 number 9)$$f(x) = x^{2} - 4x + 9$$The first step is to arrange the equation in the form $ax^{2} + bx + c$,
$$x^{2} \textcolor{#2192ff}{-\ 4}x + \textcolor{#0f0}{9}$$Identify $b$ and $c$,
$$\textcolor{#2192ff}{b} = \textcolor{#2192ff}{-4}, \ \ \textcolor{#0f0}{c} = \textcolor{#0f0}{9}$$Recall the formula for completing the square,
$$\left(x + \frac{\textcolor{#2192ff}{b}}{2}\right)^{2} - \left(\frac{\textcolor{#2192ff}{b}}{2}\right)^{2} + \textcolor{#0f0}{c}$$Substitute into the formula,
$$\left(x + \frac{\textcolor{#2192ff}{-4}}{2}\right)^{2} - \left(\frac{\textcolor{#2192ff}{-4}}{2}\right)^{2} + \textcolor{#0f0}{9}$$Simplify,
$$\left(x - 2\right)^{2} - 4 + 9$$$$\left(x - 2\right)^{2} + 5$$Therefore the final answer is,
$$f(x) = \left(x - 2\right)^{2} + 5$$

2. Express $x^{2} - 8x + 11$ in the form $(x + p)^{2} + q$ where $p$ and $q$ are constants. (9709/11/M/J/22 number 1)$$x^{2} - 8x + 11$$The first step is to arrange the equation in the form $ax^{2} + bx + c$,
$$x^{2} \textcolor{#2192ff}{-\ 8}x + \textcolor{#0f0}{11}$$Identify $b$ and $c$,
$$\textcolor{#2192ff}{b} = \textcolor{#2192ff}{-8}, \ \ \textcolor{#0f0}{c} = \textcolor{#0f0}{11}$$Recall the formula for completing the square,
$$\left(x + \frac{\textcolor{#2192ff}{b}}{2}\right)^{2} - \left(\frac{\textcolor{#2192ff}{b}}{2}\right)^{2} + \textcolor{#0f0}{c}$$Substitute into the formula,
$$\left(x + \frac{\textcolor{#2192ff}{-8}}{2}\right)^{2} - \left(\frac{\textcolor{#2192ff}{-8}}{2}\right)^{2} + \textcolor{#0f0}{11}$$Simplify,
$$\left(x - 4\right)^{2} - 16 + 11$$$$\left(x - 4\right)^{2} - 5$$Therefore the final answer is,
$$f(x) = \left(x - 4\right)^{2} - 5$$

3. The function $f$ is defined by $f(x) = 2x^{2} - 16x + 23$ for $x < 3$. Express $f(x)$ in the form $2(x + a)^{2} + b$. (9709/13/M/J/22 number 6)$$f(x) = 2x^{2} - 16x + 23$$The first step is to arrange the equation into the form $ax^{2} + bx + c$,
$$2x^{2} - 16x + 23$$Factor out the coefficient of $x^{2}$ from the first two terms,
$$2\textcolor{#2192ff}{\left(x^{2} - 8x\right)} + 23$$Complete the square for the quadratic inside parentheses,
$$2\textcolor{#2192ff}{\left[\left(x - 4\right)^{2}- \left(4\right)^{2}\right]} + 23$$Simplify inside the square brackets,
$$2\textcolor{#2192ff}{\left[\left(x - 4\right)^{2}- 16\right]} + 23$$Remove the square brackets,
$$2\left(x - 4\right)^{2} - 32 + 23$$Simplify,
$$2\left(x - 4\right)^{2} - 9$$Therefore the final answer is,
$$2\left(x - 4\right)^{2} - 9$$

4. The equation of a curve is $y = 4x^{2} + 20x + 6$. Express the equation in the form $y = a(x + b)^{2} + c$, where $a$, $b$, and $c$ are constants. (9709/12/O/N/22 number 6)$$y = 4x^{2} + 20x + 6$$The first step is to arrange the equation into the form $ax^{2} + bx + c$,
$$4x^{2} + 20x + 6$$Factor out the coefficient of $x^{2}$ from the first two terms,
$$4\textcolor{#2192ff}{\left(x^{2} + 5x\right)} + 6$$Complete the square for the quadratic inside parentheses,
$$4\textcolor{#2192ff}{\left[\left(x + \frac{5}{2}\right)^{2}- \left(\frac{5}{2}\right)^{2}\right]} + 6$$Simplify inside the square brackets,
$$4\textcolor{#2192ff}{\left[\left(x + \frac{5}{2}\right)^{2}- \frac{25}{4}\right]} + 6$$Remove the square brackets,
$$4\left(x + \frac{5}{2}\right)^{2} - 25 + 6$$Simplify,
$$4\left(x + \frac{5}{2}\right)^{2} - 19$$Therefore the final answer is,
$$4\left(x + \frac{5}{2}\right)^{2} - 19$$