Completing the Square notes for middle school

1. Completing the Square
Solving Quadratics
By
Completing the Square
Part 2
Must be a
perfect
Square

2. 2
( 5) 64
x  
5 8
x  
When you take the
square root, You
MUST consider the
Positive and
Negative answers.
5 8
x   5 8
x  
5
 5

13
x 
5
 5

3
x 
Perfect
Square
On
One side
Take
Square Root
of
BOTH SIDES
2
( 5) 64
x  

3. Perfect
Square
On
One side
Take
Square Root
of
BOTH SIDES
But what happens if you DON’T
have a perfect square on one
side…….
You make it a Perfect Square
Use the relations on next slide…

4. 2
( 6)
x   (
2 )
 
To expand a perfect square binomial:
2
12 36
x x
  
6x 2
6
We can use these relations to find the missing term….To
make it a perfect square trinomial that can be factored into a
perfect square binomial.
2
_ _
12 _
x x
 
12 2
 6
 62
6 36

36
2
x

5.  Take ½ middle term
 Then square it
The resulting trinomial is called a perfect square
trinomial,
which can be factored into a perfect square
binomial.
2
_ _
18 _ _
x x
 
18 2
 9

2
(9) 81

81 2
( 9)
x 


6. 1. 2
12 0
x x
 
1. Make one side a
perfect square
2. Add a blank to
both sides
3. Divide “b” by 2
4. Square that
answer.
5. Add it to both
sides
6. Factor 1st
side
7. Square root both
sides
8. Solve for x
2
0
x x
 
___
 ___

12 2
 6

2
(6) 36

36 36
2
( 6)
x  36
2
( 6) 36
x  
6 6
x  
6 6
x   6 6
x  
6
 6

12
x 
6
 6

0
x 
12

7. Factor this Perfect square trinomial
2
12 36
x x
 
What
is
the
Square
root
of
x
2
2
( )
x
Bring
down
sign
 6
What
is
the
Square
root
of
36
2
( 6)
x 

8. 2. 2
8 0
x x
  
1. Move constant to
other side.
2. Add a blank to
both sides
3. Divide “b” by 2
4. Square that
answer.
5. Add it to both
sides
6. Factor 1st
side
7. Square root both
sides
8. Solve for x
2
8
x x
 
___
 ___

6 2
 3

2
(3) 9

9 9
2
( 3)
x  1
2
( 3) 1
x  
3 1
x  
3 1
x   3 1
x  
3
 3

4
x 
3
 3

2
x 
6
6

9. Factor this Perfect square trinomial
2
6 9
x x
 
What
is
the
Square
root
of
x
2
2
( )
x
Bring
down
sign
 3
What
is
the
Square
root
of
9
2
( 3)
x 

10. 3. 2
8 84 0
x x
  
1. Move constant to
other side.
2. Add a blank to both
sides
3. Divide “b” by 2
4. Square that answer.
5. Add it to both sides
6. Factor 1st
side
7. Square root both
sides
8. Solve for x
2
84
x x 
___
 ___

8 2
 4

2
(4) 16

16 16
2
( 4)
x  100
2
( 4) 100
x  
4 10
x  
4 10
x   4 10
x  
4
 4

14
x 
4
 4

6
x 
8


11. Factor this Perfect square trinomial
2
8 16
x x
 
What
is
the
Square
root
of
x
2
2
( )
x
Bring
down
sign
 4
What
is
the
Square
root
of
9
2
( 4)
x 

12. 4. 2
2 15 0
x x
  
1. Move constant to
other side.
2. Add a blank to both
sides
3. Divide “b” by 2
4. Square that answer.
5. Add it to both sides
6. Factor 1st
side
7. Square root both
sides
8. Solve for x
2
15
x x 
___
 ___

2 2
 1

2
(1) 1

1 1
2
( 1)
x   16
2
( 1) 16
x  
1 4
x  
1 4
x   1 4
x  
1
 1

3
x 
1
 1

5
x 
2


13. Factor this Perfect square trinomial
2
2 1
x x
 
What
is
the
Square
root
of
x
2
2
( )
x
Bring
down
sign
 1
What
is
the
Square
root
of
9
2
( 1)
x 

14. Steps to solve Quadratics by
completing the square:
 Move the constant to side by itself.
 Make the side (w/variables) a perfect square by
adding a certain number to both sides.
 To calculate this number
– Divide “b” (middle term) by 2
– Then square that answer
 Take the square root of both sides of eq
 Then solve for x

15. In a perfect square, there is a
relationship between the coefficient of
the middle term and the constant term.
2
( 7)
x  
7 
1
(14)
2
2
7 49
2
14 49
x x
 
7