This document provides instructions for solving quadratic equations using the completing the square method. It begins with examples of perfect square trinomials and how to create them. It then walks through the steps to solve quadratic equations by completing the square, which involves getting the quadratic term alone on one side, finding the term to complete the square, factoring the resulting perfect square trinomial, and taking the square root of each side to solve for the roots. Several examples are worked through and similar problems are provided for practice.

1. Warm – Up!
 Factor the following quadratic trinomial:
1. 𝑥2
− 3𝑥 − 28 = 0
2. 3𝑥2
+ 12𝑥 − 14 = 0

2. Lesson 4: Solving
Quadratic Equations by
Completing the Square

3. What is It
 If the quadratic equation 𝑎𝑥2+𝑏𝑥+𝑐=0
is NOT factorable into two binomials,
roots can be derived by using
completing the square method. This
process involves getting the 3rd term of
the perfect square trinomial by squaring
the half of the coefficient of the middle
term, 𝑏𝑥, in symbols, 𝒂𝒙𝟐+𝒃𝒙+(
𝑏
2
)𝟐=−𝒄
+(
𝑏
)𝟐

4. What is It
 A perfect square trinomial is a
trinomial that can be written as the
square of a binomial.
 Is the result of squaring a binomial that
is the square of the first term added to
twice the product of the two terms and
the square of the last term.

5. Perfect Square Trinomials
 Examples
 x2 + 6x + 9
 x2 - 10x + 25
 x2 + 12x + 36

6. Creating a Perfect
Square Trinomial
 In the following perfect square
trinomial, the constant term is
missing.
X2 + 14x + ____
 Find the constant term by
squaring half the coefficient of
the linear term.
 (14/2)2
X2 + 14x + 49

7. Perfect Square Trinomials
 Create perfect
square trinomials.
 x2 + 20x + ___
 x2 - 4x + ___
 x2 + 5x + ___
100
4
25/4

8. Solving Quadratic Equations
by Completing the Square
Solve the following
equation by
completing the
square:
Step 1: Move
quadratic term, and
linear term to left
side of the
equation
2
8 20 0
x x
  
2
8 20
x x
 

9. Solving Quadratic Equations
by Completing the Square
Step 2: Find the term
that completes the square
on the left side of the
equation. Add that term
to both sides.
2
8 =20 +
x x
 
2
1
( ) 4 then square it, 4 16
2
8
  
2
8 20
16 16
x x
   

10. Solving Quadratic Equations
by Completing the Square
Step 3: Factor
the perfect
square trinomial
on the left side
of the equation.
Simplify the
right side of the
equation.
2
8 20
16 16
x x
   
2
( 4)( 4) 36
( 4) 36
x x
x
  
 

11. Solving Quadratic Equations
by Completing the Square
Step 4:
Take the
square
root of
each side
2
( 4) 36
x  
( 4) 6
x   

12. Solving Quadratic Equations
by Completing the Square
Step 5: Set
up the two
possibilities
and solve
4 6
4 6 and 4 6
10 and 2
x=
x
x x
x
  

 
    

13. Completing the Square-Example #2
Solve the following
equation by completing
the square:
Step 1: Move quadratic
term, and linear term to
left side of the equation,
the constant to the right
side of the equation.
2
2 7 12 0
x x
  
2
2 7 12
x x
  

14. Solving Quadratic Equations
by Completing the Square
Step 2: Find the term
that completes the square
on the left side of the
equation. Add that term
to both sides.
The quadratic coefficient
must be equal to 1 before
you complete the square, so
you must divide all terms
by the quadratic
coefficient first.
2
2
2
2 7
2
2 2 2
7 12
7
2
=-12 +
6
x x
x x
x
x
 
 
 
  
  
2
1 7 7 49
( ) then square it,
2 6
2 4 4 1
7  
  
 
 
2 49 49
16 1
7
6
2 6
x x
    

15. Solving Quadratic Equations
by Completing the Square
Step 3: Factor
the perfect
square trinomial
on the left side
of the equation.
Simplify the
right side of the
equation.
2
2
2
7
6
2
7 96 49
4 16 16
7 47
4
49 49
16 1
16
6
x x
x
x
    
 
   
 
 
 
  
 
 

16. Solving Quadratic Equations by
Completing the Square
Step 4:
Take the
square
root of
each side
2
7 47
( )
4 16
x

 
7 47
( )
4 4
7 47
4 4
7 47
4
x
i
x
i
x

  
 



17. Solving Quadratic Equations by
Completing the Square
2
2
2
2
2
1. 2 63 0
2. 8 84 0
3. 5 24 0
4. 7 13 0
5. 3 5 6 0
x x
x x
x x
x x
x x
  
  
  
  
  
Try the following examples. Do your work on your paper and then check
your answers.
 
 
1. 9,7
2.(6, 14)
3. 3,8
7 3
4.
2
5 47
5.
6
i
i



 
 
 
 
 
 
 
 
 
 