Completing the Square

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Completing the Square

• Grade 10
• Lesson 5-5

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Completing the Square

• This is an x. Show me x2.

x2
x
Show me x2 6x
3
Completing the Square
x
x
x
x
x
x
x2
x2
Let's Make a Square
4
Completing the Square
How many units are required to complete the
square?
9!
The picture is now x2 6x 9,
which factors (x3)(x3) (x3)2
5
Lets Try Another One!

• Show me x2 2x

Let's Make a Square
x2
x
x
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Completing the Square
How many units are required to complete the
square?
1!
x2
x2
The picture is now x2 2x 1
which factors (x1)(x1) (x1)2
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Last One with Manipulative
Show me x2 8x
Let's Make a Square
x
x
x
x
x
x
x
x
x2
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Completing the Square
Again, how many units are required to complete
the square?
16
So, the picture is now x2 8x 16
which factors (x4)(x4) (x4)2
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Hard One!
Complete the square for x2 18x ___ How many
units are needed?
There are not enough pieces to do this
problem. Can we do it using paper and pencil?
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What is completing the square used for?

• Completing the square is used for all those not
  factorable problems!!
• It is used to solve these equations for the
  variable.

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Rule for Completing the Square
This is now a PTS!
So, it factors into this!
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Example Find the value of c that makes this a
PTS, then write the expression as the square of
a binomial. x2-3xc

• b-3

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Example Solve by completing the square.
x26x-80

• x26x-80
• x26x8
• x26x___8___
• x26x989
• (x3)217

Dont forget Whatever you add to one side of an
equation, you MUST add to the other side!
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More Examples!

• 5x2-10x300
• x2-2x60
• x2-2x-6
• x2-2x__-6__
• x2-2x1-61
• (x-1)2-5

• 3x2-12x180
• x2-4x60
• x2-4x-6
• x2-4x__-6__
• x2-4x4-64
• (x-2)2-2

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Last Example! Write the quadratic function
yx26x16 in vertex form. What is the vertex
of the functions graph?

• yx26x16
• y-16x26x
• y-16__x26x__
• y-169x26x9
• y-7(x3)2
• y(x3)27

• If the equation, in vertex form, is y(x3)27,
  then the vertex must be (-3,7).

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Solving Quadratic Equations by Completing the
Square
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Perfect Square Trinomials

• Examples
• x2 6x 9
• x2 - 10x 25
• x2 12x 36

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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

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Perfect Square Trinomials

• Create perfect square trinomials.
• x2 20x ___
• x2 - 4x ___
• x2 5x ___

100 4 25/4
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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

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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.

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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.
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Solving Quadratic Equations by Completing the
Square

• Step 4 Take the square root of each side

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Solving Quadratic Equations by Completing the
Square

• Step 5 Set up the two possibilities and solve

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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.

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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. The quadratic coefficient must be
equal to 1 before you complete the square, so you
must divide all terms by the quadratic
coefficient first.
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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.
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Solving Quadratic Equations by Completing the
Square
Step 4 Take the square root of each side
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Solving Quadratic Equations by Completing the
Square
Try the following examples. Do your work on your
paper and then check your answers.