Completing the Square and
Finding the Vertex
Perfect Square
A polynomial that can be factored into the
following form:
2
(x + a)
Examples:
 x  2
2
x 2  10 x  25
since it factors to
 x  5
2
Completing the Square
x2 + bx + c is a perfect square if:
1 
c   b
2 
2
(The value of c will always be positive.)
Ex: Prove the following is a perfect square
x  16 x  64   x  8 x  8   x  8 
2
Half of b=-16 squared is 64=c
2
Completing the Square
Find the c that completes the square:
1. x2 + 50x + c
2. x2 – 22x + c
3. x2 + 15x + c
Factoring a Completed Square
If x2 + bx + c is a perfect square, then it will
easily factor to:
1 

 x  b
2 

2
Ex: Prove the following is a perfect square.
x  8 x  16   x  4 x  4   x  4 
2
Half of b=+8 is +4
2
Perfect Squares: Parabolas & Circles
Find the vertices of the following graphs and state
whether they are maximums or minimums.
1. y = (x + 5)2 – 5
2. y = -(x + 3)2 + 1
3. y = -3(x – 7)2 + 8
4. y = 4(x – 52)2 – 74
State the length of the radius and the coordinates of
the center for each circle below:
1. ( x – 2 )2 + ( y + 7 )2 = 64
2. x2 + y2 = 36
3. ( x + 4 )2 + ( y + 11 )2 = 5
4. ( x + 3 )2 + y2 = 175
Standard to Graphing: Quadratic
Find the vertex of the following equation by completing the square:
2
y = x + 8x + 25
GOAL
Find the “c” that
completes the square
1 

8
Plus a box, minus a box 

Complete the Square:
2


2
y = (x + 8x + 16 ) + 25 – 16
2
 4
Factor what is in the Parentheses
y = a ( x – h )2
y = (x + 4)2 + 9
+
k
Simplify
Vertex:
(-4, 9)
16
2
Standard to Graphing: Quadratic
Find the vertex of the following equation by completing the square:
y = 3x2 – 18x – 10
y=a ( x – h )
2
GOAL
+
k
y = 3(x2 – 6x + 9 ) – 10 – 3 9
y = 3(x – 3)2 – 10 – 27
y = 3(x –
3)2
9
– 37
Vertex:
1

  6 
2

2
 3 
(3,-37)
2
A new Equation?
What will the graph of the following look like:
x  4 x  y  2 y  11
2
2
Standard to Graphing: Circle
Find the center and radius of the equation by completing the square:
x2 + y2 + 6x – 12y – 9 = 0
x2 + 6x + y2 – 12y – 9 = 0
+9+9
Complete the square twice x2 + 6x + y2 – 12y = 9
Arrange similar
variables
together
Isolate the terms
with variables
(x2 + 6x + 9 ) + (y2 – 12y + 36 ) = 9 + 9 +
36
(x + 3)2 + (y – 6)2 = 54
2
1 
 6 
2 
Center:
 3
2
2
2
1

  12    6   36
2

 9
(-3, 6)
Radius:
54  9 6 3 6