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 + ___
x2 - 4x + ___
x2 + 5x + ___
100
4
25/4

5. 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
x + 8 x − 20 = 0
2
x + 8 x = 20
2

6. 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.
x + 8x +
2
W=20 +W
1
• (8) = 4 then square it, 42 = 16
2
x + 8 x + 16 = 20 + 16
2

7. 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.
x + 8 x + 16 = 20 + 16
2
( x − 4)( x − 4) = 36
( x − 4) = 36
2

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

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

10. 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 x − 7 x + 12 = 0
2
2 x − 7 x = −12
2

11. Solving Quadratic Equations
by Completing the Square
2x − 7x + W
=-12 +W
Step 2: Find the term
2x 7x
12
that completes the square
−
+ W= − + W
on the left side of the
2
2
2
2
2
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.
7
x − x +W −6 +W
=
2
2
1 7
7
• ( ) = then square it,
2 2 4
7
49
49
x − x+
= −6 +
2
16
16
2
2
49
7
 ÷ =
 4  16

12. 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.
7
49
49
x − x+
= −6 +
2
16
16
2
2
7
96 49

x− ÷ =− +
4
16 16

2
7
47

x− ÷ =−
4
16


13. Solving Quadratic Equations by
Completing the Square
Step 4:
Take the
square
root of
each side
7 2
−47
(x − ) =
4
16
7
−47
(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. x + 2 x − 63 = 0
2
2. x + 8 x −84 = 0
2
3. x − 5 x − 24 = 0
2
4. x + 7 x +13 = 0
2
5. 3 x 2 +5 x + 6 = 0
1. ( −9, 7 )
2.(6, −14)
3. ( −3,8 )
 −7 ± i 3 
4. 
÷

÷
2


 −5 ± i 47 
5. 
÷

÷
6

