Completing the Square Technique

Due to math content, this page has special requirements (including JavaScript) for full functionality.
With your current viewing scenario, it is not appearing and behaving as it is supposed to!
Please visit Dr. Carol J.V. Fisher's Homepage to learn what this site has to offer.
Watch the "Welcome" video to get started—hope to see you back here soon!

Dr. Carol J.V. Fisher's Homepage

For this exercise, you need ♥ INTERNET EXPLORER 6.0 and above, with MathPlayer installed.♥

QUADRATIC FUNCTIONS and the
COMPLETING THE SQUARE TECHNIQUE

Jump right to the exercises!

DEFINITION (quadratic function)
A quadratic function is a function that can be written in the form y= ax2 +bx+c for a≠0 ;
this is called the standard form of the quadratic function.
Every quadratic function can also be written in the form y=a (x-h) 2 +k for a≠0 ;
this is called the vertex form of the quadratic function.

A quadratic function must have an x2 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.

A quadratic function is renamed from standard form to vertex form using the technique of completing the square, which is discussed next.
Every quadratic function graphs as a parabola.

DEFINITION (perfect square trinomial)
A perfect square trinomial is a trinomial that can be written in the form (x+k )2 .

All of the following are perfect square trinomials:
x2+ 2x+1= (x+1)2
x2- 6x+9= (x-3)2
x2+ 45x +425 =(x+ 25) 2

An investigation of (x+k)2 =x2+ 2kx+k2 shows that there is a very simple relationship
between the coefficient of the x term (2k)
and the constant term (k2) 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!

COMPLETING THE SQUARE TECHNIQUE
The process of finding the correct number to add to an expression of the form x2 +bx to form a perfect square trinomial is called completing the square.
The correct number to add is (b 2)2 ;
that is, take the coefficient of the x term, divide it by 2, and then square the result.
Then, x2 +bx+( b2 )2 =( x+b2 )2 .

EXAMPLE (completing the square)
What number must be added to x2 -3x to form a perfect square trinomial?
Then, what perfect square trinomial results?
SOLUTION:
The number to add is: (- 32) 2=9 4
Notice that the number inside the parentheses, - 32 ,
is the number that goes next to x in the perfect square:
x2 -3x+ ( -3 2 ) 2 = x2 -3x+ 94 = (x-3 2)2 .

EXAMPLE (renaming as an expression involving a perfect square trinomial)
Rename x2 +6x as an expression involving a perfect square trinomial.
SOLUTION: Note that (6 2)2 =9 .
x2 +6x

=x2 +6x+9-9	(add zero in an appropriate form)
=(x2 +6x+9)-9	(regroup)
=(x+ 3)2 -9	(rename the perfect square)

The technique of completing the square only works when the coefficient of the x2 term is 1 .
Any other coefficient must be factored out before completing the square, as shown next.

GOING FROM STANDARD FORM TO VERTEX FORM
In what follows, the completing the square technique is applied to the standard form, ax2 +bx+c ,
to change it to vertex form, a (x-h) 2 +k .

ax2 +bx+c

= (ax 2+bx) +c	(group first two terms)
=a(x 2+b ax) +c	(factor a≠0 out of the first two terms)
=a(x 2+b ax+ (b2 a)2 -(b 2a) 2)+c	(add zero in an appropriate form inside the parentheses; note that ba ÷2= ba⋅ 12= b2a )
=a(x 2+b ax+ (b2 a)2 )-a( b2a )2+ c	(distributive law)
=a(x +b2 a)2 +stuff	(rename as a perfect square)

Notice that the x-value of the vertex is - b 2a !
This is worth recording and memorizing:

VERTEX OF A QUADRATIC FUNCTION IN STANDARD FORM
The vertex of the parabola y= ax2 +bx+c ( a≠0 )
has x-value equal to - b 2a .

EXAMPLE (writing a quadratic function in vertex form)
Write y=5x 2+3 x-1 in vertex form and give the coordinates of the vertex.

FIRST SOLUTION:
In this first approach, the technique of completing the square is used to put the function in vertex form,
from which the coordinates of the vertex are read.

y=5x2 +3x-1
    =5(x 2+3 5x+ (310 )2 -(3 10) 2)-1
    =5(x +310 )2- 5⋅9 100-1
    =5(x +310 )2- 29 20
Thus, the vertex is (-3 10,- 2920) .

SECOND SOLUTION:
In this second approach, the vertex formula is used to find the x-value of the vertex;
then the corresponding y-value is found;
then, this information is used to write the vertex form.

From the vertex formula, the x-value of the vertex is:  - b 2a=- 32(5) =-3 10
The corresponding y-value is:   5(- 310) 2+3 (-310 )-1= - 2920
The vertex form, y=a (x-h) 2 +k , corresponding to
h=-3 10 ,    k=-20 29 ,    and    a=5 ,   is   y=5(x +310 )2- 29 20 .

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.

QUADRATIC FUNCTIONS and the COMPLETING THE SQUARE TECHNIQUE

QUADRATIC FUNCTIONS and the
COMPLETING THE SQUARE TECHNIQUE