6. a2 ± 2ab + b2
• Perfect Square Trinomials are
trinomials of the form a2 ± 2ab +
b2, which can be expressed as
squares of binomials.
• When Perfect Square Trinomials
are factored, the factored form is
(a ± b)2
7. Knowing the
previous
information will
help us when
Completing the
Square
It is very important to understand how to Complete the Square as you
will be using this method in other modules!
10. Completing the Square
Completing the Square in another way to Factor a Quadratic
Equation.
“Take Half and Square” are words you hear when
referencing “Completing the Square”
11. Completing the Square
Completing the Square in another way to Factor a Quadratic
Equation.
“Take Half and Square” are words you hear when
referencing “Completing the Square”
EOC Note:
When a problem says “to solve”, “find the x-intercepts” or
the equation is set = 0,
then you will Factor using any Factoring method that
you have learned.
13. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.
You can probably look at this problem and know what the answer will be, BUT let’s
Factor using Completing the Square!
14. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.
You can probably look at this problem and know what the answer will be, BUT let’s
Factor using Completing the Square!
Step 1: Move the +9 to the other side by subtracting (leave
spaces as shown)
x2 + 6x + _____ = -9 + ______
15. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.
You can probably look at this problem and know what the answer will be, BUT let’s
Factor using Completing the Square!
Step 1: Move the +9 to the other side by subtracting (leave
spaces as shown)
x2 + 6x + _____ = -9 + ______
Step 2: “Take half and Square” the coefficient of the linear
term, which is 6.
Take half of 6 which is 3, then square 3, which is
9.
16. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.
You can probably look at this problem and know what the answer will be, BUT let’s
Factor using Completing the Square!
Step 1: Move the +9 to the other side by subtracting (leave
spaces as shown)
x2 + 6x + _____ = -9 + ______
Step 2: “Take half and Square” the coefficient of the linear
term, which is 6.
Take half of 6 which is 3, then square 3, which is
9.
17. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.
You can probably look at this problem and know what the answer will be, BUT let’s
Factor using Completing the Square!
Step 1: Move the +9 to the other side by subtracting (leave
spaces as shown)
x2 + 6x + _____ = -9 + ______
Step 2: “Take half and Square” the coefficient of the linear
term, which is 6.
Take half of 6 which is 3, then square 3, which is
9.
Step 3: Add that 9 to both sides (and place where the
squares are)—This step is legal because we are adding the same number
to both sides.
x2 + 6x + 9 = -9 + 9
18. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.
You can probably look at this problem and know what the answer will be, BUT let’s
Factor using Completing the Square!
Step 1: Move the +9 to the other side by subtracting (leave
spaces as shown)
x2 + 6x + _____ = -9 + ______
Step 2: “Take half and Square” the coefficient of the linear
term, which is 6.
Take half of 6 which is 3, then square 3, which is
9.
Step 3: Add that 9 to both sides (and place where the
squares are)—This step is legal because we are adding the same number
to both sides.
x2 + 6x + 9 = -9 + 9
19. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.
You can probably look at this problem and know what the answer will be, BUT let’s
Factor using Completing the Square!
Step 1: Move the +9 to the other side by subtracting (leave
spaces as shown)
x2 + 6x + _____ = -9 + ______
Step 2: “Take half and Square” the coefficient of the linear
term, which is 6.
Take half of 6 which is 3, then square 3, which is
9.
Step 3: Add that 9 to both sides (and place where the
squares are)—This step is legal because we are adding the same number
to both sides.
x2 + 6x + 9 = -9 + 9
Step 4: Factor the left side of the equation and simplify the
right side.
(x + 3)2 = 0
20. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.
You can probably look at this problem and know what the answer will be, BUT let’s
Factor using Completing the Square!
Step 1: Move the +9 to the other side by subtracting (leave
spaces as shown)
x2 + 6x + _____ = -9 + ______
Step 2: “Take half and Square” the coefficient of the linear
term, which is 6.
Take half of 6 which is 3, then square 3, which is
9.
Step 3: Add that 9 to both sides (and place where the
squares are)—This step is legal because we are adding the same number
to both sides.
x2 + 6x + 9 = -9 + 9
Step 4: Factor the left side of the equation and simplify the
right side.
(x + 3)2 = 0
Step 5: Take the Square Root of both sides, then solve for x.
21. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.
You can probably look at this problem and know what the answer will be, BUT let’s
Factor using Completing the Square!
Step 1: Move the +9 to the other side by subtracting (leave
spaces as shown)
x2 + 6x + _____ = -9 + ______
Step 2: “Take half and Square” the coefficient of the linear
term, which is 6.
Take half of 6 which is 3, then square 3, which is
9.
Step 3: Add that 9 to both sides (and place where the
squares are)—This step is legal because we are adding the same number
to both sides.
x2 + 6x + 9 = -9 + 9
Step 4: Factor the left side of the equation and simplify the
right side.
(x + 3)2 = 0
Step 5: Take the Square Root of both sides, then solve for x.
Step 6: Solve for x: x + 3 = 0 --> x = -3
23. Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square.
Step 1: Move the +4 to the other side (by subtracting 4).
x2 - 8x + _____ = -4 + _____
24. Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square.
Step 1: Move the +4 to the other side (by subtracting 4).
x2 - 8x + _____ = -4 + _____
25. Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square.
Step 1: Move the +4 to the other side (by subtracting 4).
x2 - 8x + _____ = -4 + _____
Step 2: “Take half and Square” the coefficient of the linear
term, which is -8.
Take half and square
26. Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square.
Step 1: Move the +4 to the other side (by subtracting 4).
x2 - 8x + _____ = -4 + _____
Step 2: “Take half and Square” the coefficient of the linear
term, which is -8.
Take half and square
Step 3: Add 16 to both side ( and place where the squares
are).
x2 - 8x + 16 = -4 + 16
27. Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square.
Step 1: Move the +4 to the other side (by subtracting 4).
x2 - 8x + _____ = -4 + _____
Step 2: “Take half and Square” the coefficient of the linear
term, which is -8.
Take half and square
Step 3: Add 16 to both side ( and place where the squares
are).
x2 - 8x + 16 = -4 + 16
Step 4: Factor the left side and simplify the right side.
(x - 4)2 = 12
28. Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square.
Step 1: Move the +4 to the other side (by subtracting 4).
x2 - 8x + _____ = -4 + _____
Step 2: “Take half and Square” the coefficient of the linear
term, which is -8.
Take half and square
Step 3: Add 16 to both side ( and place where the squares
are).
x2 - 8x + 16 = -4 + 16
Step 4: Factor the left side and simplify the right side.
(x - 4)2 = 12
Step 5: Take the square root of both sides. x – 4 =
29. Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square.
Step 1: Move the +4 to the other side (by subtracting 4).
x2 - 8x + _____ = -4 + _____
Step 2: “Take half and Square” the coefficient of the linear
term, which is -8.
Take half and square
Step 3: Add 16 to both side ( and place where the squares
are).
x2 - 8x + 16 = -4 + 16
Step 4: Factor the left side and simplify the right side.
(x - 4)2 = 12
Step 5: Take the square root of both sides. x – 4 =
Step 6: Solve for x x =
30. Example 3: Factor 4x2 - 2x + 3 = 0 using Completing the Square.
********There is a new step because the coefficient of x2 is not 1.
31. Example 3: Factor 4x2 - 2x + 3 = 0 using Completing the Square.
********There is a new step because the coefficient of x2 is not 1.
32. Example 3: Factor 4x2 - 2x + 3 = 0 using Completing the Square.
********There is a new step because the coefficient of x2 is not 1.
Notice how
we divided
4x2 – 2x + 3 = 0 x2 – x+ by 4!
=0
33. Example 3: Factor 4x2 - 2x + 3 = 0 using Completing the Square.
********There is a new step because the coefficient of x2 is not 1.
Notice how
we divided
4x2 – 2x + 3 = 0 x2 – x+ by 4!
=0
Step 1: Move the + to the other side by subtracting .
x2 - x + ___ = - + ___
34. Example 3: Factor 4x2 - 2x + 3 = 0 using Completing the Square.
********There is a new step because the coefficient of x2 is not 1.
Notice how
we divided
4x2 – 2x + 3 = 0 x2 – x+ by 4!
=0
Step 1: Move the + to the other side by subtracting .
x2 - x + ___ = - + ___
Step 2: “Take half and Square” the coefficient of the linear
term,
, which becomes .
35. Example 3: Factor 4x2 - 2x + 3 = 0 using Completing the Square.
********There is a new step because the coefficient of x2 is not 1.
Notice how
we divided
4x2 – 2x + 3 = 0 x2 – x+ by 4!
=0
Step 1: Move the + to the other side by subtracting .
x2 - x + ___ = - + ___
Step 2: “Take half and Square” the coefficient of the linear
term,
, which becomes .
Step 3: Add to both sides
Go to the next slide…
36.
37. Step 4: Factor the left side of the equation and simplify the
right
side.
38. Step 4: Factor the left side of the equation and simplify the
right
side.
Step 5: Take the square root of both sides.
39. Step 4: Factor the left side of the equation and simplify the
right
side.
Step 5: Take the square root of both sides.
Step 6: Solve for x:
or
40. Step 4: Factor the left side of the equation and simplify the
right
side.
Step 5: Take the square root of both sides.
Step 6: Solve for x:
or
What is “half” of the following numbers?
1.½ ½ times ½ ¼
2.¼ ¼ times ½ ⅛
3.⅓ ⅓ times ½
4.⅜ ⅜ times ½
41. Very Nice Site for
Interactive
Examples of
Completing the
Square!
42. Very Nice Site for
Interactive
Examples of
Completing the
Square!
43. Let’s review a few things…
• Let’s suppose your answer looked like the following—
• Do you see something else that we could do to simplify
this equation?
• There are a few more steps. First we need to clean up
the .
• Go to the next slide to see the steps…
45. • Our old equation was
• Our new equation is
• Now there is another “no no”. We need to rationalize the
denominator in order to get rid of the radical in the
denominator.
• Now our new equation is
46. • Now, let’s solve:
• Add 4 to both sides. Final
answer is:
47. Links
Practice Practice Practice
Problems Problems Problems
Video:
Explanation Practice
Example
Video:
Examples Examples
a=1
