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Completing    the  Square
Table of Contents Slide 3-5: Perfect Square Trinomials Slide 7: Completing the Square Slides 8-11: Examples Slide 13-1...
What are Perfect Square        Trinomials?• Let’s begin by simplifying a few  binomials.• Simplify each.1.(x + 4)22.(x – 7...
1. (x + 4)2 = x2 + 4x + 4x + 16 x2 + 8x   + 162. (x – 7)2   =   x2 – 7x – 7x + 49 x2 – 14x +   493. (2x + 1)2 = 4x2 + 2x...
a2 ± 2ab + b2• Perfect Square Trinomials are  trinomials of the form a2 ± 2ab +  b2, which can be expressed as  squares of...
Knowing the           previous      information will        help us when       Completing the            SquareIt is very ...
Completing the Square
Completing the SquareCompleting the Square in another way to Factor a Quadratic                          Equation.
Completing the SquareCompleting the Square in another way to Factor a Quadratic                          Equation.     “Ta...
Completing the SquareCompleting the Square in another way to Factor a Quadratic                          Equation.     “Ta...
Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.
Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.  You can probably look at this problem and know what the an...
Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.  You can probably look at this problem and know what the an...
Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.  You can probably look at this problem and know what the an...
Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.  You can probably look at this problem and know what the an...
Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.  You can probably look at this problem and know what the an...
Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.  You can probably look at this problem and know what the an...
Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.  You can probably look at this problem and know what the an...
Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.  You can probably look at this problem and know what the an...
Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.  You can probably look at this problem and know what the an...
Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square.
Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square. Step 1: Move the +4 to the other side (by subtracting 4).  ...
Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square. Step 1: Move the +4 to the other side (by subtracting 4).  ...
Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square. Step 1: Move the +4 to the other side (by subtracting 4).  ...
Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square. Step 1: Move the +4 to the other side (by subtracting 4).  ...
Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square. Step 1: Move the +4 to the other side (by subtracting 4).  ...
Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square. Step 1: Move the +4 to the other side (by subtracting 4).  ...
Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square. Step 1: Move the +4 to the other side (by subtracting 4).  ...
Example 3: Factor 4x2 - 2x + 3 = 0 using Completing the Square.********There is a new step because the coefficient of x2 i...
Example 3: Factor 4x2 - 2x + 3 = 0 using Completing the Square.********There is a new step because the coefficient of x2 i...
Example 3: Factor 4x2 - 2x + 3 = 0 using Completing the Square.********There is a new step because the coefficient of x2 i...
Example 3: Factor 4x2 - 2x + 3 = 0 using Completing the Square.********There is a new step because the coefficient of x2 i...
Example 3: Factor 4x2 - 2x + 3 = 0 using Completing the Square.********There is a new step because the coefficient of x2 i...
Example 3: Factor 4x2 - 2x + 3 = 0 using Completing the Square.********There is a new step because the coefficient of x2 i...
Step 4: Factor the left side of the equation and simplify the     rightside.
Step 4: Factor the left side of the equation and simplify the     rightside.Step 5: Take the square root of both sides.
Step 4: Factor the left side of the equation and simplify the     rightside.Step 5: Take the square root of both sides.Ste...
Step 4: Factor the left side of the equation and simplify the     rightside.Step 5: Take the square root of both sides.Ste...
Very Nice Site for   Interactive  Examples of Completing the    Square!
Very Nice Site for   Interactive  Examples of Completing the    Square!
Let’s review a few things…•   Let’s suppose your answer looked like the following—•   Do you see something else that we co...
Let’s simplify theSince
•   Our old equation was•   Our new equation is•   Now there is another “no no”. We need to rationalize the    denominator...
• Now, let’s solve:• Add 4 to both sides. Final  answer is:
LinksPractice      Practice   PracticeProblems      Problems   Problems               Video:Explanation              Pract...
Notes   completing the square
Notes   completing the square
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Notes completing the square

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Notes completing the square

  1. 1. Completing the Square
  2. 2. Table of Contents Slide 3-5: Perfect Square Trinomials Slide 7: Completing the Square Slides 8-11: Examples Slide 13-16: Simplifying Answers Audio/Video and Interactive Sites Slide 12: Video/Interactive Slide 17: Videos/Interactive
  3. 3. What are Perfect Square Trinomials?• Let’s begin by simplifying a few binomials.• Simplify each.1.(x + 4)22.(x – 7)23.(2x + 1)24.(3x – 4)2
  4. 4. 1. (x + 4)2 = x2 + 4x + 4x + 16 x2 + 8x + 162. (x – 7)2 = x2 – 7x – 7x + 49 x2 – 14x + 493. (2x + 1)2 = 4x2 + 2x + 2x + 1  4x2 + 4x + 1
  5. 5. a2 ± 2ab + b2• Perfect Square Trinomials are trinomials of the form a2 ± 2ab + b2, which can be expressed as squares of binomials.• When Perfect Square Trinomials are factored, the factored form is (a ± b)2
  6. 6. Knowing the previous information will help us when Completing the SquareIt is very important to understand how to Complete the Square as you will be using this method in other modules!
  7. 7. Completing the Square
  8. 8. Completing the SquareCompleting the Square in another way to Factor a Quadratic Equation.
  9. 9. Completing the SquareCompleting the Square in another way to Factor a Quadratic Equation. “Take Half and Square” are words you hear when referencing “Completing the Square”
  10. 10. Completing the SquareCompleting the Square in another way to Factor a Quadratic Equation. “Take Half and Square” are words you hear when referencing “Completing the Square” EOC Note:When a problem says “to solve”, “find the x-intercepts” or the equation is set = 0, then you will Factor using any Factoring method that you have learned.
  11. 11. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.
  12. 12. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square. You can probably look at this problem and know what the answer will be, BUT let’s Factor using Completing the Square!
  13. 13. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square. You can probably look at this problem and know what the answer will be, BUT let’s Factor using Completing the Square! Step 1: Move the +9 to the other side by subtracting (leave spaces as shown) x2 + 6x + _____ = -9 + ______
  14. 14. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square. You can probably look at this problem and know what the answer will be, BUT let’s Factor using Completing the Square! Step 1: Move the +9 to the other side by subtracting (leave spaces as shown) x2 + 6x + _____ = -9 + ______ Step 2: “Take half and Square” the coefficient of the linear term, which is 6. Take half of 6 which is 3, then square 3, which is 9.
  15. 15. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square. You can probably look at this problem and know what the answer will be, BUT let’s Factor using Completing the Square! Step 1: Move the +9 to the other side by subtracting (leave spaces as shown) x2 + 6x + _____ = -9 + ______ Step 2: “Take half and Square” the coefficient of the linear term, which is 6. Take half of 6 which is 3, then square 3, which is 9.
  16. 16. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square. You can probably look at this problem and know what the answer will be, BUT let’s Factor using Completing the Square! Step 1: Move the +9 to the other side by subtracting (leave spaces as shown) x2 + 6x + _____ = -9 + ______ Step 2: “Take half and Square” the coefficient of the linear term, which is 6. Take half of 6 which is 3, then square 3, which is 9. Step 3: Add that 9 to both sides (and place where the squares are)—This step is legal because we are adding the same number to both sides. x2 + 6x + 9 = -9 + 9
  17. 17. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square. You can probably look at this problem and know what the answer will be, BUT let’s Factor using Completing the Square! Step 1: Move the +9 to the other side by subtracting (leave spaces as shown) x2 + 6x + _____ = -9 + ______ Step 2: “Take half and Square” the coefficient of the linear term, which is 6. Take half of 6 which is 3, then square 3, which is 9. Step 3: Add that 9 to both sides (and place where the squares are)—This step is legal because we are adding the same number to both sides. x2 + 6x + 9 = -9 + 9
  18. 18. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square. You can probably look at this problem and know what the answer will be, BUT let’s Factor using Completing the Square! Step 1: Move the +9 to the other side by subtracting (leave spaces as shown) x2 + 6x + _____ = -9 + ______ Step 2: “Take half and Square” the coefficient of the linear term, which is 6. Take half of 6 which is 3, then square 3, which is 9. Step 3: Add that 9 to both sides (and place where the squares are)—This step is legal because we are adding the same number to both sides. x2 + 6x + 9 = -9 + 9 Step 4: Factor the left side of the equation and simplify the right side. (x + 3)2 = 0
  19. 19. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square. You can probably look at this problem and know what the answer will be, BUT let’s Factor using Completing the Square! Step 1: Move the +9 to the other side by subtracting (leave spaces as shown) x2 + 6x + _____ = -9 + ______ Step 2: “Take half and Square” the coefficient of the linear term, which is 6. Take half of 6 which is 3, then square 3, which is 9. Step 3: Add that 9 to both sides (and place where the squares are)—This step is legal because we are adding the same number to both sides. x2 + 6x + 9 = -9 + 9 Step 4: Factor the left side of the equation and simplify the right side. (x + 3)2 = 0 Step 5: Take the Square Root of both sides, then solve for x.
  20. 20. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square. You can probably look at this problem and know what the answer will be, BUT let’s Factor using Completing the Square! Step 1: Move the +9 to the other side by subtracting (leave spaces as shown) x2 + 6x + _____ = -9 + ______ Step 2: “Take half and Square” the coefficient of the linear term, which is 6. Take half of 6 which is 3, then square 3, which is 9. Step 3: Add that 9 to both sides (and place where the squares are)—This step is legal because we are adding the same number to both sides. x2 + 6x + 9 = -9 + 9 Step 4: Factor the left side of the equation and simplify the right side. (x + 3)2 = 0 Step 5: Take the Square Root of both sides, then solve for x. Step 6: Solve for x: x + 3 = 0 --> x = -3
  21. 21. Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square.
  22. 22. Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square. Step 1: Move the +4 to the other side (by subtracting 4). x2 - 8x + _____ = -4 + _____
  23. 23. Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square. Step 1: Move the +4 to the other side (by subtracting 4). x2 - 8x + _____ = -4 + _____
  24. 24. Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square. Step 1: Move the +4 to the other side (by subtracting 4). x2 - 8x + _____ = -4 + _____ Step 2: “Take half and Square” the coefficient of the linear term, which is -8. Take half and square
  25. 25. Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square. Step 1: Move the +4 to the other side (by subtracting 4). x2 - 8x + _____ = -4 + _____ Step 2: “Take half and Square” the coefficient of the linear term, which is -8. Take half and square Step 3: Add 16 to both side ( and place where the squares are). x2 - 8x + 16 = -4 + 16
  26. 26. Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square. Step 1: Move the +4 to the other side (by subtracting 4). x2 - 8x + _____ = -4 + _____ Step 2: “Take half and Square” the coefficient of the linear term, which is -8. Take half and square Step 3: Add 16 to both side ( and place where the squares are). x2 - 8x + 16 = -4 + 16 Step 4: Factor the left side and simplify the right side. (x - 4)2 = 12
  27. 27. Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square. Step 1: Move the +4 to the other side (by subtracting 4). x2 - 8x + _____ = -4 + _____ Step 2: “Take half and Square” the coefficient of the linear term, which is -8. Take half and square Step 3: Add 16 to both side ( and place where the squares are). x2 - 8x + 16 = -4 + 16 Step 4: Factor the left side and simplify the right side. (x - 4)2 = 12 Step 5: Take the square root of both sides. x – 4 =
  28. 28. Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square. Step 1: Move the +4 to the other side (by subtracting 4). x2 - 8x + _____ = -4 + _____ Step 2: “Take half and Square” the coefficient of the linear term, which is -8. Take half and square Step 3: Add 16 to both side ( and place where the squares are). x2 - 8x + 16 = -4 + 16 Step 4: Factor the left side and simplify the right side. (x - 4)2 = 12 Step 5: Take the square root of both sides. x – 4 = Step 6: Solve for x x =
  29. 29. Example 3: Factor 4x2 - 2x + 3 = 0 using Completing the Square.********There is a new step because the coefficient of x2 is not 1.
  30. 30. Example 3: Factor 4x2 - 2x + 3 = 0 using Completing the Square.********There is a new step because the coefficient of x2 is not 1.
  31. 31. Example 3: Factor 4x2 - 2x + 3 = 0 using Completing the Square.********There is a new step because the coefficient of x2 is not 1. Notice how we divided 4x2 – 2x + 3 = 0 x2 – x+ by 4! =0
  32. 32. Example 3: Factor 4x2 - 2x + 3 = 0 using Completing the Square.********There is a new step because the coefficient of x2 is not 1. Notice how we divided 4x2 – 2x + 3 = 0 x2 – x+ by 4! =0 Step 1: Move the + to the other side by subtracting . x2 - x + ___ = - + ___
  33. 33. Example 3: Factor 4x2 - 2x + 3 = 0 using Completing the Square.********There is a new step because the coefficient of x2 is not 1. Notice how we divided 4x2 – 2x + 3 = 0 x2 – x+ by 4! =0 Step 1: Move the + to the other side by subtracting . x2 - x + ___ = - + ___ Step 2: “Take half and Square” the coefficient of the linear term, , which becomes .
  34. 34. Example 3: Factor 4x2 - 2x + 3 = 0 using Completing the Square.********There is a new step because the coefficient of x2 is not 1. Notice how we divided 4x2 – 2x + 3 = 0 x2 – x+ by 4! =0 Step 1: Move the + to the other side by subtracting . x2 - x + ___ = - + ___ Step 2: “Take half and Square” the coefficient of the linear term, , which becomes . Step 3: Add to both sides Go to the next slide…
  35. 35. Step 4: Factor the left side of the equation and simplify the rightside.
  36. 36. Step 4: Factor the left side of the equation and simplify the rightside.Step 5: Take the square root of both sides.
  37. 37. Step 4: Factor the left side of the equation and simplify the rightside.Step 5: Take the square root of both sides.Step 6: Solve for x: or
  38. 38. Step 4: Factor the left side of the equation and simplify the rightside.Step 5: Take the square root of both sides.Step 6: Solve for x: or What is “half” of the following numbers? 1.½  ½ times ½  ¼ 2.¼ ¼ times ½  ⅛ 3.⅓  ⅓ times ½  4.⅜  ⅜ times ½ 
  39. 39. Very Nice Site for Interactive Examples of Completing the Square!
  40. 40. Very Nice Site for Interactive Examples of Completing the Square!
  41. 41. Let’s review a few things…• Let’s suppose your answer looked like the following—• Do you see something else that we could do to simplify this equation?• There are a few more steps. First we need to clean upthe .• Go to the next slide to see the steps…
  42. 42. Let’s simplify theSince
  43. 43. • Our old equation was• Our new equation is• Now there is another “no no”. We need to rationalize the denominator in order to get rid of the radical in the denominator.• Now our new equation is
  44. 44. • Now, let’s solve:• Add 4 to both sides. Final answer is:
  45. 45. LinksPractice Practice PracticeProblems Problems Problems Video:Explanation Practice Example Video:Examples Examples a=1

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