Question

3.2.4 Journal: Completing the Square Journal Algebra I Sem 2 Name: Date: Scenario: The Stage Instructions: View the video found on page 1 of this Journal activity. Using the information provided in the video, answer the questions below. Show your work for all calculations. Analyze The Students' Conjectures Malcolm wants to build an outdoor stage with a total area of 350 square feet. The length of the stage should be 3 feet shorter than the width. He calculated the equation to be w2 − 3w = 350 1. Complete the table to summarize each student's suggestion for figuring out the equation: (2 points: 1 point for each row of the chart) Student Conjecture Malcolm's bandmate Malcolm 2. Malcolm and his bandmate have different ideas for figuring out the equation. Which one do you think will make it easiest to solve the equation? Why? (2 points) 3. Malcolm's bandmate starts by making a factor table for w2 – 3w – 350 = 0. He is looking for the factors (w + p)(w + q) = 0, where p • q = –350 and p + q = –3. Fill in the last row of the table with different factors, p and q, so that p • q = –350. (2 points). p q p + q 10 –35 –25 –10 35 25 50 –7 43 –50 7 –43 _____ _____ _____ 4. The bandmate's factor table is not complete. It does not contain all the factors of -350. Does the factoring table contain the factors that can be used to solve w2 – 3w – 350 = 0? Explain your reasoning. (1 point) 5. To make a perfect square trinomial, Malcolm said the rule for figuring out the number to add is . Is he correct? If not, what is the rule? (1 point) 6. Calculate the number that you need to add to each side of the equation w2 – 3w = 350 to create a perfect square trinomial. Add this number to each side of the equation. Show your work. (3 points) 7. Factor the trinomial. (3 points) 8. Now that you've factored the equation, find the square root of each side and solve for w. Show your work and both solutions. (2 points) 9. Do both of these solutions make sense in terms of t

Answer 1

1. | Malcolm's bandmate | Factor the expression w² - 3w - 350 = 0 |

2. Malcolm's bandmate's method of factoring the expression will make it easier to solve the equation.

3. Factor the Expression

| -14 | 25 | 11 |

4. The bandmate's factor table is not complete. It does not contain all the factors of -350.

Completing the Square

5. No, Malcolm is not correct.

6. The equation is (w - 1.5)² = 352.25

7. The trinomial is 20.25 or w = -17.25

8. The value of w = 20.25 or w = -17.25

9. Yes, both of these solutions make sense in terms of the problem.

1. | Student | Conjecture |

|---|---|

| Malcolm | Use the quadratic formula: w = (−b ± √(b² - 4ac)) / 2a |

| Malcolm's bandmate | Factor the expression w² - 3w - 350 = 0 |

2. Malcolm's bandmate's method of factoring the expression will make it easier to solve the equation. This is because factoring the expression will allow us to isolate w directly. The quadratic formula, on the other hand, involves more complex calculations.

3. Factor the Expression

| p | q | p + q |

|---|---|---|

| 10 | -35 | -25 |

| -10 | 35 | 25 |

| 50 | -7 | 43 |

| -50 | 7 | -43 |

| 14 | -25 | -11 |

| -14 | 25 | 11 |

4. The bandmate's factor table is not complete. It does not contain all the factors of -350. This is because the table only includes factors that add up to -3. However, there are other factors of -350 that add up to a different number. For example, 14 and -25 are factors of -350, and they add up to -11. These factors can also be used to solve the equation w² - 3w - 350 = 0.

Completing the Square

5. No, Malcolm is not correct. The rule for figuring out the number to add is to take half of the coefficient of the x term, square it, and add it to both sides of the equation.

6. To complete the square, we need to take half of the coefficient of the x term, square it, and add it to both sides of the equation. The coefficient of the x term is -3, so half of it is -1.5. Squaring -1.5 gives us 2.25. Therefore, we need to add 2.25 to both sides of the equation.

w² - 3w = 350

w² - 3w + 2.25 = 350 + 2.25

(w - 1.5)² = 352.25

7. Factor the trinomial:

(w - 1.5)² = 352.25

w - 1.5 = ±√352.25

w - 1.5 = ±18.75

w = 20.25 or w = -17.25

8. Solve for w:

w = 20.25 or w = -17.25

9. Yes, both of these solutions make sense in terms of the problem. Malcolm wants to build an outdoor stage with a total area of 350 square feet. The length of the stage should be 3 feet shorter than the width. If the width is 20.25 feet, then the length is 17.25 feet. If the width is -17.25 feet, then the length is -20.25 feet. However, this solution does not make sense in terms of the real world, as we cannot have a negative length. Therefore, the only solution that makes sense is w = 20.25.