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- 1. Completing the Square
- 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. 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. 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. 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. 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. Completing the Square
- 8. Completing the SquareCompleting the Square in another way to Factor a Quadratic Equation.
- 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. 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. Example 1: Factor x2 + 6x + 9 = 0 using Completing the Square.
- 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. 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. 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. 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. 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. 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. 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. 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. 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. Example 2: Factor x2 - 8x + 4 = 0 using Completing the Square.
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Step 4: Factor the left side of the equation and simplify the rightside.
- 36. Step 4: Factor the left side of the equation and simplify the rightside.Step 5: Take the square root of both sides.
- 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. 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. Very Nice Site for Interactive Examples of Completing the Square!
- 40. Very Nice Site for Interactive Examples of Completing the Square!
- 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. Let’s simplify theSince
- 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. • Now, let’s solve:• Add 4 to both sides. Final answer is:
- 45. LinksPractice Practice PracticeProblems Problems Problems Video:Explanation Practice Example Video:Examples Examples a=1

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