Completing the square

Basic functions: Polynomial functions

Completing the square

Completing the square is a useful method for computing the vertex and zeros of a quadratic function. Finding zeros boils down to the reduction of a quadratic equation of the form

$ax^2+bx+c=0$ to an equation of the form

$a(x+p)^2+q=0$ for some real numbers $a$ , $b$ , $c$ , $p$ , and $q$ with $a\neq 0$ .

Once a quadratic function is in the form $a(x+p)^2+q$ , then the coordinates of the vertex are equal to $(-p,q)$ .

GeoGebra

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An example explains the method.

Example of completing the square In order to solve the equation

$x^2+6x-7=0$ we rewrite it as

$x^2+6x+9=16$ because the left-hand side is now a square, namely $(x+3)^2$ . Indeed

$x^2+6x+9=(x+3)^2$ The established equation can be solved by the basic techniques:

$x+3=4\quad\text{or}\quad x+3=-4$ and so the solution is

$x=1\quad\text{or}\quad x=-7$

The method in the example above works in general for the quadratic equation $ax^2+bx+c=0$ : if the coefficient of $x^2$ is equal to 1, then $p=1$ . Otherwise, you divide the equation first by $a$ to get a leading coefficient that is equal to $1$ . Next, take half of the coefficient of $x$ to make a square on the left-hand side, that is, choose this numerical value for $q$ . In other words, set $p=\frac{b}{2}$ . Determine the constant on the right-hand side of the equation, in other words, calculate $-q=\frac{b^2}{4}-c$ . If the constant $-q$ is negative, then there is no solution. If this constant is equal to zero, then there is exactly one solution, namely $x=-q$ . If this constant is positive, then there are two solutions.

If $a\neq 0$ and $a\neq 1$ , then you can determine the formula for completing the square: every quadratic function

$f(x)=ax^2+bx+c$ can be written as

$f(x)=a(x+p)^2+q$ with

$p=-\frac{b}{2a},\quad q=-\frac{D}{4a}\quad\text{and}\quad D=b^2-4ac$ The expression $D=b^2-4ac$ is called the discriminant of the quadratic function $f(x)=ax^2+bx+c$ .

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We give other examples of completing the square, also with a coefficient of $x^2$ unequal to one.

Calculate exactly the real solution of the following quadratic equation by completing the square.

$x^2-4x+4=0$

$x=2$

We write down the steps in completing the square

$\begin{aligned} x^2-4x+4=0 & \phantom{abcxyz} \blue{\text{the given quadratic equation}} \\ \\ \left(x-2\right)^2-\left(-2\right)^2+4=0 &\phantom{abcxyz} \blue{\text{completing the square}}\\ \\ \left(x-2\right)^2+0=0& \phantom{abcxyz} \blue{\text{simplification}}\\ \\ \left(x-2\right)^2=0& \phantom{abcxyz} \blue{\text{isolation of the square}}\\ \\ x-2=0 & \blue{\phantom{abcxyz}\text{one solution}}\\ \\ x=2 & \blue{\phantom{abcxyz}\text{final answer}} \end{aligned}$

New example

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Mathcentre videos

Completing the Square - Animation (1:50)

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Completing the Square by Inspection (19:38)