Completing the Square

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Module 11 Topic 1

• Completing
• the
• Square

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• Table of Contents
• Slide 3-5 Perfect Square Trinomials
• Slide 7 Completing the Square
• Slides 8-11 Examples
• Slide 13-16 Simplifying Answers
• Audio/Video and Interactive Sites
• Slide 12 Video/Interactive
• Slide 17 Videos/Interactive

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What are Perfect Square Trinomials?

• Lets begin by simplifying a few binomials.
• Simplify each.
• (x 4)2
• (x 7)2
• (2x 1)2
• (3x 4)2

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1. (x 4)2 x2 4x 4x 16 ?x2 8x 16
2. (x 7)2 x2 7x 7x 49 ?x2 14x 49
3. (2x 1)2 4x2 2x 2x 1 ? 4x2 4x 1
4. (3x 4)2 9x2 12x 12 16 ? 9x2 24x
   16

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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

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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!
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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
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.
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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 lets 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 --gt
x -3
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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
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Example 3 Factor 4x2 - 2x 3 0 using
Completing the Square. There is a new
step because the coefficient of x2 is not 1.
New step If the coefficient does ? 1, then
you must divide everything by that coefficient.
Notice how we divided by 4!
4x2 2x 3 0? x2 x 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
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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?
• ½ ? ½ times ½ ? ¼
• ¼? ¼ times ½ ? ?
• ? ? ? times ½ ?
• ? ? ? times ½ ?

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Lets review a few things

• Lets 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
  

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Lets simplify the
Since
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• 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

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• Now, lets solve
• Add 4 to both sides. Final answer is