18.03 Completing the square (Extended)

Completing the square is the process of rewriting a quadratic expression from the expanded form $x^2+2bx+c$x2+2bx+c into the square form $\left(x+b\right)^2+c-b^2$(x+b)2+c−b2. The name comes from how the method was first performed by ancient Greek mathematicians who were actually looking for numbers to complete a physical square.

There are two ways to think about the completing the square method: visually (using a square), and algebraically. We will see that we can use this method to solve and factorise quadratics that previous methods could not, leading us to the quadratic formula in our next lesson.

 

Visual method - drawing a box

Consider the following quadratic expression:

$x^2+4x-5$x2+4x−5

To complete the square the steps involved are:

• Move the constant term to the end
• Create a box and complete the square
• Keep the expression balanced using subtraction

As this is a visual method, it's best to learn by watching the process in action, so here is an example.

Worked example

example 1

(a) Write $x^2+4x-5$x2+4x−5 in the form $\left(x+m\right)^2+n$(x+m)2+n by completing the square.

Think: Here is a visual of the method:

Do: From the video we can see that  $x^2+4x-5=\left(x+2\right)^2-9$x2+4x−5=(x+2)2−9.

(b) Hence, solve the equation $x^2+4x-5=0$x2+4x−5=0 after completing the square.

Think: We can now solve the equation with algebraic manipulation.

$x^2+4x-5$x2+4x−5	$=$=	$0$0
$\left(x+2\right)^2-9$(x+2)2−9	$=$=	$0$0	Completed square from (a)
$\left(x+2\right)^2$(x+2)2	$=$=	$9$9	Take the constant to the RHS
$x+2=3$x+2=3	or	$x+2=-3$x+2=−3	Take square root both sides
$x=1$x=1	or	$x=-5$x=−5	Solve for the $x$x values

Algebraic method

Let's now look at how we can complete the square using algebraic steps.

Completing the square algebraically:

• Factor out the coefficient of the $x^2$x2 term, unless it is $1$1
• Halve the coefficient of $x$x and square it, add and subtract this value
• Factorise the terms that now form a perfect square
• Combine the constant and term left over from completing the square

 

Worked examples

Example 2

Write the expression $x^2-6x+5$x2−6x+5 in the form $\left(x-m\right)^2+n$(x−m)2+n by completing the square.

Think: check that the coefficient of $x^2$x2 is $1$1 so no factorisation is required. Halve the coefficient of the $x$x term, add and subtract the coefficient squared.

Do: 

$x^2-6x+5$x2−6x+5	$=$=	$x^2-6x+\left(-3\right)^2-\left(-3\right)^2+5$x2−6x+(−3)2−(−3)2+5	Half of $-6$−6 is $-3$−3.
	$=$=	$x^2-6x+9-9+5$x2−6x+9−9+5
	$=$=	$\left(x^2-6x+9\right)-4$(x2−6x+9)−4	Put the first three terms in a bracket
	$=$=	$\left(x-3\right)^2-4$(x−3)2−4	Factorise the bracket as perfect square

Example 3

Solve the equation $3x^2+6x-8=0$3x2+6x−8=0 using the completion of the square method.

Think: Check the coefficient of $x^2$x2. It is not $1$1, therefore factor out the $3$3 so that $x^2$x2 has a coefficient of $1$1 when inside the bracket. 

Do: 

$0$0	$=$=	$3x^2+6x-8$3x2+6x−8
$0$0	$=$=	$3\left(x^2+2x-\frac{8}{3}\right)$3(x2+2x−83​)	Take the coefficient of $x^2$x2 out as factor
$0$0	$=$=	$3\left(x^2+2x+1^2-1^2-\frac{8}{3}\right)$3(x2+2x+12−12−83​)	Add and subtract the square of half the coefficient of $x$x
$0$0	$=$=	$3\left[\left(x+1\right)^2-1-\frac{8}{3}\right]$3[(x+1)2−1−83​]	Factorise the first three terms as a perfect square
$0$0	$=$=	$3\left[\left(x+1\right)^2-\frac{11}{3}\right]$3[(x+1)2−113​]	Combine the constant terms
$0$0	$=$=	$\left(x+1\right)^2-\frac{11}{3}$(x+1)2−113​	Divide both sides by $3$3
$\frac{11}{3}$113​	$=$=	$\left(x+1\right)^2$(x+1)2	Take constant to the other side
$\pm\sqrt{\frac{11}{3}}$±√113​	$=$=	$x+1$x+1	Take the square root of both sides
$x$x	$=$=	$\sqrt{\frac{11}{3}}-1$√113​−1 or $-\sqrt{\frac{11}{3}}-1$−√113​−1	Subtract $1$1 from both answers leaving answer in exact form

Reflect: We could have simplified the working for this question if we divided both sides by $3$3 earlier.

Watch this video for an interactive that will help you to visualise the process. Watch video.

 

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5