The document discusses solving quadratic equations by completing the square. It provides examples of perfect square trinomials and how to find the missing constant term to create them. The steps for solving a quadratic equation by completing the square are described: 1) move all terms to the left side, 2) find and add the "completing the square" term, 3) factor into a perfect square trinomial, 4) take the square root of each side, 5) solve the resulting equations. Additional examples demonstrate applying these steps to solve quadratic equations algebraically.

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

