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Completing the Square: From Standard to Vertex Form

Discover how completing the square transforms quadratic equations from standard to vertex form. Learn the process step by step with examples, including changing coefficients. Practice with provided equations and understand the significance of finding the vertex in parabolas.

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Completing the Square: From Standard to Vertex Form

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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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