55 Completing the Square part one

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5-5 Completing the Squarepart one

• Main Ideas
• Solve quadratic equations by completing the
  square.

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Using Algebra Tiles to Complete the Square

• Question Given b, what is the value of c that
  makes x2 bx c a perfect square trinomial?
• 1. Use algebra tiles to model the expression x2
  6x

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• 2. Arrange the tiles in a square. Your
  arrangement will be incomplete in one corner.

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• 3. Determine the number of 1 tiles needed to
  complete the square.
• By adding nine 1-tiles, you
• can see that
• x2 6x 9 (x 3)2

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Copy and complete the table below by following
the previous steps.
1
x2 2x 1 (x 1)2
4
x2 4x 4 (x 2)2
16
x2 8x 16 (x 4)2
25
x2 10x 25 (x 5)2
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Drawing Conclusions

• Look for patterns in the last column of your
  table. Consider the general statement
• x2 bx c (x d)2.
• A. How is d related to b in each case?
• d ½ b
• B. How is c related to d in each case?
• c d2
• C. How can you obtain the numbers in the second
  column of the table directly from the
  coefficients of x in the expressions from the
  first column?
• Find the square of half of the coefficient.

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Completing the Square
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Example 1

• Find the value of c that makes x2 12x c a
  perfect square. Then write the trinomial as a
  perfect square.
• x2 12x c
• b 12
• x2 12x 36
• (x 6)2

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Example 2a1 equations

• Solve the equation by completing the square.
• x2 8x 20 0
• x2 8x 20
• b 8
• x2 8x 20
• x2 8x 16 20 16
• x2 8x 16 36
• (x 4)2 36

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Example 3a ? 1 equations (Divide thru)

• Solve the equation by completing the square.
• 2x2 5x 3 0
• 2x2 5x -3

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Example 4Equations with Complex Solutions

• Solve the equation by completing the square.
• x2 4x 11 0
• x2 4x -11

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Converting to the Vertex Formof a Quadratic

• Write the quadratic function y x2 6x 16 in
  vertex form, y a(x h)2 k. What is the
  vertex of the functions graph?
• y x2 6x 16
• y __ (x2 6x __) 16
• y 9 (x 3)2 16
• y (x 3)2 7
• y a(x h)2 k
• Vertex(-3, 7)