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Completing the Square

Completing the Square. What do you get when you foil the following expressions?. (x + 1) (x+1)=. (x + 6) 2 =. (x + 7) 2 =. (x + 2) (x+2) =. (x + 8) 2 =. (x + 3) (x+3) =. (x + 4) (x+4) =. (x + 9) 2 =. (x + 5) (x+5) =. (x + 10) 2 =.

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Completing the Square

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  1. Completing the Square

  2. What do you get when you foil the following expressions? (x + 1) (x+1)= (x + 6)2 = (x + 7)2 = (x + 2) (x+2)= (x + 8)2 = (x + 3) (x+3)= (x + 4) (x+4)= (x + 9)2 = (x + 5) (x+5)= (x + 10)2 =

  3. What do you get when you foil the following expressions? (x + 1)2 = x2 + 2x + 1 (x + 10)2 = x2 + 20x + 100 (x + 2)2 = x2 + 4x + 4 (x - 13)2 = x2 - 26x + 169 (x - 3)2 = x2 - 6x + 9 (x - 25)2 = x2 - 50x + 625 (x - 4)2 = x2 - 8x + 16 (x – 0.5)2 = x2 - x + 0.25 x2 – 6.4x + 10.24 (x + 5)2 = x2 + 10x + 25 (x – 3.2)2 =

  4. Fill in the missing number to complete a perfect square. x2 + 2x + ____ x2 - 14x + ___ x2 + 8x + ___ x2 – 20x + ___ x2 + 6x + ___ x2 + 16x + _____

  5. Fill in the missing number to complete a perfect square. x2 + 10x + ___ x2 + 10x + 25 = (x + 5)2 x2 - 30x + ___ x2 - 30x + 225 = (x - 15)2 x2 – 2.8x + ___ x2 – 2.8x + 1.96 = (x – 1.4)2 x2 + 18x + ___ x2 + 18x + 81 = (x + 9)2 x2 + 12x + ___ x2 + 12x + 36 = (x + 6)2 x2 + 0.5x + _____ x2 + 0.5x + 0.0625 = (x – 0.25)2

  6. The vertex is at (-7, -59) Changing from standard form to vertex form By completing the square on a quadratic in standard form, it is changed into vertex form Change to vertex form: y = x2 + 14x - 10 y = x2 + 14x + ____ - 10 y = x2 + 14x + 49 - 10 - 49 y = (x + 7)2 -59

  7. The vertex is at (6, -31) Changing from standard form to vertex form By completing the square on a quadratic in standard form, it is changed into vertex form Change to vertex form: y = x2 - 12x + 5 y = x2 - 12x + ____ + 5 y = x2 - 12x + 36 + 5 - 36 y = (x - 6)2 - 31

  8. The vertex is at (14, 4) Changing from standard form to vertex form By completing the square on a quadratic in standard form, it is changed into vertex form Change to vertex form: y = x2 - 28x + 200 y = x2 - 28x + ____ + 200 y = x2 - 28x + 196 + 200 - 196 y = (x - 14)2 + 4

  9. The vertex is at (0.375, -1.140625) Changing from standard form to vertex form By completing the square on a quadratic in standard form, it is changed into vertex form Change to vertex form: y = x2 – 0.75x - 1 y = x2 – 0.75x + ____ + - 1 y = x2 – 0.75x + .140625 - 1 - .140625 y = (x – 0.375)2 – 1.140625

  10. Change to vertex form: y = x2 + 4x + 10 y = x2 + 4x + ___ + 10 y = x2 + 4x + 4 + 10 - 4 y = (x + 2)2 + 6

  11. Change to vertex form: y = x2 + 19x - 1 y = x2 + 19x + ___ - 1 y = x2 + 19x + 90.25 - 1 – 90.25 y = (x + 9.5)2 - 91.25

  12. If the leading coefficient is not equal to 1, completing the square is slightly more difficult. More Complicated Versions of Completing the Square Directions for Completing the Square: 1.) Move the constant out of the way. 2.) Factor out A from the x2 and x term. 3.) Determine what is half of the remaining B. 4.) Square it and put this in for C. 5.) Put in a constant to cancel out the last step. 6.) Write the parenthesis as a perfect square and simplify everything else.

  13. Vertex at (-1, 8) Change to vertex form: y = 2x2 + 4x + 10 y = 2(x2 + 2x + ___) + 10 - ___ y = 2(x2 + 2x + 1) + 10 - 2 y = 2(x + 1)2 + 8

  14. Vertex at (-2, 10) Change to vertex form: y = 3x2 + 12x + 22 y = 3(x2 + 4x + ___) + 22 - ___ y = 3(x2 + 4x + 4) + 22 - 12 y = 3(x + 2)2 + 10

  15. Change to vertex form: y = 6x2 - 48x + 65

  16. Change to vertex form: y = 7x2 - 98x + 400

  17. Change to vertex form: y = 12x2 - 60x + 312

  18. Vertex at (2, -12) Change to vertex form: y = -5x2 + 20x - 32 y = -5(x2 - 4x + ___) - 32 - ___ y = -5(x2 - 4x + 4) - 32 + 20 y = -5(x - 2)2 - 12

  19. Vertex at (6, 163) Change to vertex form: y = -6x2 + 72x - 53 y = -6(x2 - 12x + ___) - 53 - ___ y = -6(x2 - 12x + 36) - 53 + 216 y = -6(x - 6)2 + 163

  20. Methods of Locating the Vertex of a Parabola: If the quadratic is in vertex form: The vertex is @ (h, k): If the quadratic is in factored form: The x value of the vertex is halfway between the roots. Plug in & solve to find the y value. If the quadratic is in standard form: Complete the square to change to vertex form.

  21. Vertex at (-0.3, -2.45) Change to vertex form:

  22. Change to vertex form:

  23. Change to vertex form:

  24. Change to vertex form:

  25. Solve by completing the square.

  26. Solve by completing the square.

  27. Example: Solve by completing the square: x2 + 6x – 8 = 0 x2 + 6x - 8 = 0 x2 + 6x = 8 x2 + 6x + ___= 8 + ___ x2 + 6x + 9 = 8 + 9 (x+3)2 = 17

  28. Solve by completing the square:

  29. Solve by completing the square:

  30. Solve by completing the square: This is called the Quadratic Formula. You must memorize it!!!

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