Completing the Square

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10.5

• Completing the Square

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10.5 Completing the Square

• Goals / I can
• Solve quadratic equations by completing the square

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10.5 Completing the Square

• Review
• Remember weve solved quadratics using 3
  different ways
• Graphing
• Square Roots
• Factoring

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10.5 Completing the Square
How many solutions are there? What are they?
y x2 4x 5 Solutions are -1 and 5
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10.5 Completing the Square
Use the Square Root method to solve
1. 25x2 16
2. 9m2 100
3. 49b2 64 0
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10.5 Completing the Square

• Example 1
• x2 2x 24 0
• (x 4)(x 6) 0
• x 4 0 x 6 0
• x 4 x 6

Example 2 x2 8x 11 0 x2 8x 11 is
prime therefore, another method must be used to
solve this equation.
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10.5 Completing the Square

• The easiest trinomials to look at are often
  perfect squares because they always have the SAME
  characteristics.

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10.5 Completing the Square

• x 8x 16 is factored into
• (x 4) notice that the 4 is (½ 8)

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2
2
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10.5 Completing the Square

• This is ALWAYS the case with perfect squares.
  The last term in the binomial can be found by the
  formula ½ b
• Using this idea, we can make polynomials that
  arent perfect squares into perfect squares.

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10.5 Completing the Square

• Example
• x 22x ____ What number
• would fit in the
• last term to make
• it a perfect
• square?

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10.5 Completing the Square
2

• (½ 22) 121
• SO.. x 22x 121 should be a
• perfect square.
• (x 11)

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2
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10.5 Completing the Square

• What numbers should be added to each equation to
  complete the square?
• x 20x
• x - 8x
• x 50x

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2
2
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10.5 Completing the Square

• This method will work to solve ALL quadratic
  equations
• HOWEVER
• it is messy to solve quadratic equations by
  completing the square if a ? 1 and/or b is an odd
  number.
• Completing the square is a GREAT choice for
  solving quadratic equations if a 1 and b is an
  even number.

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10.5 Completing the Square
Example 2 a ? 1, b is not even 3x2 5x 2 0

• Example 1
• a 1, b is even
• x2 6x - 7 0
• x2 6x 9 7 9
• (x 3)2 16
• x 3 4
• x 7 OR 1

OR
x 1 OR x ?
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10.5 Completing the Square
2

• Solving x bx c
• x 8x 48 I want to solve
• this using perfect
• squares.
• How can I make the left side of the equation a
  perfect square?

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10.5 Completing the Square
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2

• Use ½ b (½ 8) 16
• Add 16 to both sides of the equation. (we MUST
  keep the equation equivalent)
• x 8x 16 48 16
• Make the left side a perfect square binomial.
• (x 4) 64

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10.5 Completing the Square

• x 4 8
• SO.
• x 4 8 x 4 -8
• x 4 x -12

-
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10.5 Completing the Square
2

• Solving x bx c 0
• x 12x 11 0 Since it is not a
• perfect square,
• move the 11 to
• the other side.
• x 12x -11 Now, can you
• complete the square
• on the left side?

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10.5 Completing the Square
Find the value of c that makes the expression a
perfect square trinomial. Then write the
expression as the square of a binomial.
1. x2 8x c
2. x2 ? 12x c
3. x2 3x c
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10.5 Completing the Square
Solve x2 16x 15 by completing the square.
SOLUTION
x2 16x 15
Write original equation.
x2 16x ( 8)2 15 ( 8)2
(x 8)2 15 ( 8)2
Write left side as the square of a binomial.
(x 8)2 49
Simplify the right side.
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10.5 Completing the Square
x 8 7
Take square roots of each side.
x 8 7
Add 8 to each side.
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10.5 Completing the Square
2

• x 12x ? -11 ?
• x 12x -11
• (x )

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2
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10.5 Completing the Square

• Complete the square
• x - 20x 32 0

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10.5 Completing the Square

• Complete the square
• x 3x 5 0

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10.5 Completing the Square

• Complete the square
• x 9x 136

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10.5 Completing the Square

• Still a little foggy?
• If so, watch this video to see if it will help