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Math-3 Lesson 2-3
Completing the Square
Different forms of the square function.
Vertex form (“transformation form”)
cbxaxxf 2)(
khxaxf 2)()(
Standard (polynomial) form
Intercept Form (x-intercept)
))(()( qxpxaxf
5)4(3)( 2 xxf
1572)( 2 xxxf
)4)(3(2)( xxxf
Perfect squares What do all of the following numbers have in common?
They are all the square of a number.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
255
164
93
42
11
2
2
2
2
2
10010
819
648
497
366
2
2
2
2
2
)1)(1( xx
)2)(2( xx
)3)(3( xx
Convert the vertex form quadratic into a standard form quadratic..
2)1( xy
2)2( xy
2)3( xy
122 xx
442 xx
962 xx
“Right times right is right, right plus right is middle”
Do you notice a pattern with the coefficients of ‘x’?
2)( axy ))(( axax 22 2 aaxx
“Right times right is right, right plus right is middle”
Look at the constant term
2)( ax ))(( axax 22 2 aax
It equals the square of half the middle term.
2
2 2*2
1
aa
Perfect Square Trinomial
12)1( 22 xxx
khxay 2)(
The square of a binomial
vertex form
)1( 2 xy
Where is the vertex?
0 )1( 2 xy
(-1, 0)
Can you recognize a perfect square trinomial? 22 )1(12)( xxxxf The square of a binomial
34)( 2 xxxf
44)( 2 xxxf
65)( 2 xxxf
86)( 2 xxxf
96)( 2 xxxf
)3)(1( xx No
)2)(2( xx Yes 2)2( x
)3)(2( xx No
)4)(2( xx No
)3)(3( xx Yes 2)3( x
22 )1(12)( xxxxf The square of a binomial
44)( 2 xxxf
96)( 2 xxxf
2)2( x
2)3( x
How do you recognize them?
What number squared = 1 and added to itself = 2?
(Notice that any number added to itself
is the same as two times itself)
A + A = 2A
cbxxxf 2)(
cA 2
2bA
bA 2
What number squared = 4 and added to itself = 4?
What number squared = 9 and added to itself = 6?
What number squared = “c” and 2 times itself = “b”?
cb 2
2
Standard Form Quadratic
cbxaxy 2
Perfect Square Trinomial
2
2
21
bbxxy
Interpret the following equation in set-builder notation.
}2 1, 0, 1,x,){()( 2 hxxf
Where are the vertices of perfect square trinomials?
What is special about a Perfect square Trinomial?
- Binomial squared.
- One x-intercept.
- Vertex is x-intercept.
- Constant term is a
perfect square.
- Constant term is
(½ the coefficient of x)²
2
2
2
bbxxy
Graph
Equation
cbxxy 2
khxy 2)(
Convert standard form to vertex form
(by completing the square)
Completing the Square: converting
into a perfect square trinomial,
then converting it to the square of a binomial.
Vocabulary
dbxx 2
2
2
2
bbxx
22
22
bd
bx
db
2
2
khxy 2)(
1st we need to figure out what number to add to “Complete the Square”
2x
x4 4
4
2xx
x
x8
8
x44
x
x
16
2
2
8
x
x42x
2x x8
Your turn:
They are formed by squaring a binomial.
There is only one x-intercept.
The x-intercept is the vertex of the parabola.
The right-most term is the middle term divided by two
then squared
Tell 4 things you know about
perfect square trinomials.
2
2
2
bbxxy
Completing the Square
2x
x5
2
105
2xx
x
x10
10
x55
x
x
25
25
x
x52x
2x x10
Your turn:
2x
x6
6
6
2xx
x
x12
-12
x66
x
x
2)6(
36
x
x62x
2x x12“complete the square”
What must the number ‘c’
be equal to for it to be a
“perfect square trinomial”?
cxx 22C = ?
cxx 142C = ?
C = ? cxx 162
2
2
2
1
2
2
14
49
2
2
16
64
What is the
equivalent
binomial squared?
22 )1(12 xxx
22 )7(4914 xxx
22 )8(6416 xxx
Your turn:
64162 xxy
49142 xxy
2)8( xy
Rewrite the equation as the
square of a binomial:
2)7( xy
442 xxy
36122 xxy2)6( xy
2)2( xy
What form (std, vertex, intercept) would you call the
binomial squared?
2)1( xy
All these vertex form equation have their
vertexes ON the x-axis.
2)7( xy2)6( xy
2)2( xy
We can write vertex form for parabolas
whose vertex is NOT on the x-axis.
4)2( 2 xy 4)3( 2 xy
Converting a standard form into a vertex form.
1262 xxy
khxay 2)(
What number would
complete the square? 2
2
6
939
Rewrite 12 so that 9 is
one of the addends. 3962 xxy
3962 xxy Notice the perfect
square trinomial!!!
Convert the perfect square trinomial
to the square of a binomial 3)3( 2 xy
Vertex form!!!!!!
Converting a standard form into a vertex form.
1262 xxy
khxay 2)(
What number is needed
to complete the square?
2
2
6
9
An alternative way is
add 9 and subtract 9. 129962 xxy
)129(962 xxyNotice the perfect
square trinomial!!!
Convert the perfect square trinomial
to the square of a binomial 3)3( 2 xy
Vertex form!!!!!!
Converting a standard form into a vertex form.
742 xxy
khxay 2)(
What number is needed
to complete the square?
2
2
4
4
Add 4, subtract 4 74442 xxy
)74(442 xxyNotice the perfect
square trinomial!!!
Convert to the square of a binomial 11)2( 2 xy
Vertex form!!!!!!
Your turn:
1782 xxy
khxay 2)(
What number would
complete the square?
2
2
8
16
Add 16, subtract 16. 17161682 xxy
)1716(1682 xxyNotice the perfect
square trinomial!!!
Convert to the square of a binomial 33)4( 2 xy
Vertex form!!!!!!
Rewrite in vertex form by
completing the square.
Your turn:
20102 xxy
khxay 2)(
What number would
complete the square?
2
2
10
25
Add and subtract 25 202525102 xxy
4525102 xxy Notice the perfect
square trinomial!!!
Convert to the square of a binomial 45)5( 2 xy
Vertex form!!!!!!
Rewrite in vertex form by
completing the square.
Perfect Square Trinomial
cbxaxy 2
khxay 2)(
Complete the square
Turn standard form into vertex
form, then solve directly.
(by extracting a square root)
(by completing the square)
x intercepts
162 xxy
52162 xxy
8)3( 2 xy
12)8( 2 xy
Your turn: Rewrite in vertex form by
completing the square.
Find ‘zeroes’ by completing the square
(1) Convert to vertex form: 362 xxy
39962 xxy
(2) Set y = 0
(3) Isolate the square, undo the square
2)3(12 x
312 x
123x
123x
2
2
6
9
12)3( 2 xy
12)3(0 2 x2)3(12 x
123x
Solving by completing the square (1) Convert to vertex form: 4102 xxy
42525102 xxy
(2) Set y = 0
(3) Isolate the square, undo the square
2)5(29 x
529 x
295x
295x
2
2
10
25
29)5( 2 xy
29)5(0 2 x2)5(29 x
295x
622 xxy
1362 xxy
71 ,71 x
iix 23 ,23
Your turn: Solve by completing the square.
