This document provides instructions for solving quadratic equations by completing the square. It explains the steps: 1) organize the equation terms, 2) add the term to complete the square, 3) factor the perfect square trinomial, 4) take the square root of each side, 5) solve for x. Examples are provided and readers are instructed to work through additional examples on their own.

1. Solving Quadratic
Equations by
Completing the Square

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

3. 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

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

5. Solving Quadratic Equations
by Completing the Square
Begin with an equation in standard polynomial form: Ax 2 + Bx + C =
0
Example: x + 8 x − 20 = 0
2
Step 1: Move quadratic
term, and linear term
to left side of the x + 8 x = 20
2
equation (put the
constant by itself on
the right side of the
equation.

6. Solving Quadratic Equations
by Completing the Square
Step 2: Find the term
that completes the square
x + 8x +
2
W=20 +W
on the left side of the 1
equation. Add that term • (8) = 4 then square it, 42 = 16
to both sides. 2
This term will always be the square of (B/2)!
x + 8 x + 16 = 20 + 16
2
CAUTION: A must equal 1 before beginning
step 2! If A is not equal to 1, divide all terms by
A.

7. Solving Quadratic Equations
by Completing the Square
Step 3: Factor
the perfect
square trinomial
on the left side
x + 8 x + 16 = 20 + 16
2
of the equation.
Simplify the
right side of the
( x − 4)( x − 4) = 36
equation.
( x − 4) = 36
2
The factors on the left side will ALWAYS be (x – B/2)(x+B/2)!

8. Solving Quadratic Equations
by Completing the Square
Step 4:
Take the ( x + 4) = 36
2
square
root of ( x + 4) = ±6
each side

9. Solving Quadratic Equations
by Completing the Square
Step 5: Set x = −4 ± 6
up the two
possibilities x = −4 − 6 and x = −4 + 6
and solve x = −10 and x=2

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

11. Solving Quadratic Equations
by Completing the Square
2x − 7x + W2
=-12 +W
Step 2: Find the term
+ W= − + W
2
that completes the square 2x 7x 12
−
on the left side of the 2 2 2
equation. Add that term
x − x +W −6 +W
7
to both sides. 2
=
2
Remember, the quadratic 2
coefficient (A) must be 1 7 7 7 49
• ( ) = then square it,  ÷ =
equal to 1 before you begin 2 2 4  4  16
step 2, if not you must
divide all terms by the
quadratic coefficient to 7 49 49
make A = 1.
x − x+
2
= −6 +
2 16 16

12. Solving Quadratic Equations
by Completing the Square
Step 3: 7 49 49
Factor x − x+
2
= −6 +
the perfect 2 16 16
square trinomial
2
on the left side
 7 96 49
of the equation.
x− ÷ =− +
Simplify the
right side of the
 4 16 16
equation. 2
 7 47
x− ÷ =−
 4 16

13. Solving Quadratic Equations by
Completing the Square
Step 4: 7 2 −47
Take the (x − ) =
square
4 16
root of 7 −47
each side (x − ) = ±
4 4
7 i 47
x= ±
4 4
7 ± i 47
x=
4

14. Solving Quadratic Equations by
Completing the Square
Try the following examples. Do your work on your paper and then check
your answers.
1. ( −9, 7 )
1. x + 2 x − 63 = 0
2
2.(6, −14)
2. x + 8 x −84 = 0
2
3. ( −3,8 )
3. x − 5 x − 24 = 0
2
 −7 ± i 3 
4. x + 7 x +13 = 0
2
4.  ÷
 2 ÷
5. 3 x 2 +5 x + 6 = 0  
 −5 ± i 47 
5. 
 ÷
÷
 6 