Unit 6: Say It with Symbols

Transcription

1 Unit 6: Say It with Symbols I can solve linear and quadratic equations using symbolic reasoning. A problem often requires finding solutions to equations. In previous Units, you developed strategies for solving linear and quadratic equations. In this Investigation, you will use the properties of real numbers to extend these strategies. Lesson 1: Solving Linear Equations I can solve linear and quadratic equations using symbolic reasoning The steps below show one way to solve x = x for x. How could you explain Steps 1, 3, and 5, in the solution? The solution begins by subtracting 4x from each side of the equation. Could you begin with a different first step? Explain.

2 How can you check that x = 25 is the correct solution? The preceding examples use the of that you learned in the Unit Moving Straight Ahead. You can or the same quantity from each side of an equation to write an equation. You can or each side of an equation by the same quantity to write an equation. You can use these properties as well as the and properties to solve equations. Problem 3.1 A. The school choir is selling boxes of greeting cards to raise money for a trip. The equation for the profit in dollars P in terms of the number of boxes sold s is P = 5s ( s) 1. What information do the expressions 5s and s represent in the situation? What information do 100 and 2s represent? 2. Use the equation to find the number of boxes the choir must sell to make a $200 profit. Explain.

3 3. How many boxes must the choir sell to break even (income = expenses)? Explain. 4. Write a simpler expression for profit. Explain how your expression is equivalent to the original expression for profit. 5. One of the choir members wrote the following expression for profit: 5s 2(50 + s). Explain whether this expression is equivalent to the original expression for profit. B. Describe how to solve an equation that has parentheses such as 200 = 5s ( s) without using a table or graph. C. Solve each equation for x when y = 0. Check your solutions. 1. Y = 5 + 2(3 + 4x) 3. Y = 5 + 2(3 4x) 2. Y = 5 2(3 + 4x) 4. Y = 5 2(3 4x)

4 Lesson 2: Solving More Linear Equations I can solve linear and quadratic equations using symbolic reasoning Ms. Lucero wants to install tiles around her square swimming pool. She finds the following two advertisements for tile companies. The equations below shows the estimated costs C (in dollars) of buying and installing N border tiles. Cover and Surround it: C C = 1, (N 12) Tile and Beyond: C T = (N 10) You can use to show different uses for a variable: C C means for ; C T means for. Do the equations make sense, given the description above for each company s chargers? Explain. Is the cost of Tile and Beyond always cheaper than the cost of Cover and Surround It? Explain. Ms. Lucero wants to know when the costs of each company were equal. How can Ms. Lucero use the equation C C = C T to answer her question?

5 Problem 3.2 A. 1. Without using a table or graph, find the number of tiles for which the two costs are equal. 2. How can you check that your solution is correct? 3. How can you use a graph or table to find the number of tiles for which the two costs are equal? 4. For what numbers of tiles is Tile and Beyond cheaper than Cover and Surround It (C T < C c )? B. Use the strategies that you developed in Problem 3.1 and in Question A to solve each equation for x. Check your solutions. 1. 3x = 5 + 2(3 + 4x)

6 2. 3x = 5 2(3 + 4x) x = 2(3 + 4x) (1 x) = 5 2(3 4x) C. For each pair of equations, o Find the values of x that makes y 1 = y 2 without using a table or graph. o State whether the linear equation y 1 = y 2 has a finite number of solutions, an infinite number of solutions, or no solutions. o Graph the pair of equations. 1. y 1 = 3(2x 5) and y 2 = 2(3x 1) + x 2. y 1 = 3(2x 5) and y 2 = 2(3x 1) + 7

7 3. y 1 = 3(2x 5) and y 2 = 2(3x 1) - 13 Practice ACE Questions: # 8-17 Lesson 3: Factoring Quadratic Equations I can solve linear and quadratic equations using symbolic reasoning Sometimes mathematical problems that appear to be different are actually the same. Finding the x- intercepts of the graph of y = x 2 + 5x is the same as solving the equation x 2 + 5x = 0. The to x 2 + 5x = 0 are also called the of the equation. In Frogs, Fleas, and Painted Cubes, you found the solutions or roots by using a table or graph of y = x 2 + 5x as shown. What is the factored form of x 2 + 5x?

8 What is the relationship between the factored form of x 2 + 5x and the x-intercepts of the graph of y = x 2 + 5x? Explain. To factor the expression x 2 + 5x + 6, Trevor draws the area model shown. Does the model represent x 2 + 5x + 6? What are the factors of x 2 + 5x + 6? What are the x-intercepts of the graph of x 2 + 5x + 6? What is the relationship between the x-intercepts of the graph of y = x 2 + 5x + 6 and the factored form of x 2 + 5x + 6? Algebra provides important tools, such as factoring, that can help solve quadratic equations such as x 2 + 5x = 0 without using tables or graphs. Before using this tool, you need to review how to write quadratic expressions in factored form. Problem 3.3 A. Jakai suggests the method below to factor x 2 + 8x Use an area model to show why Jakai s method works for the expression x 2 + 8x + 12.

9 2. Could Jakai have used another factor pair, such as 1 and 12 or 3 and 4, to make an area model for the expression x 2 + 8x + 12? Explain. B. Use a method similar to Jakai s to write each expression in factored form. Show why each factored form is correct. C. 1. Examine the following expressions. How are they similar to and different from those in Question B? 2. Will Jakai s method for factoring work on these expressions? If so, use his method to write them in factored form. If not, find another way to write each in factored form.

10 D. 1. Examine the following expressions. How are they similar to and different from those in Question B? 2. Will Jakai s method work on these expressions? If so, write them in factored form. If not, find another way to write each in factored form. Explain why your expression is equivalent to the original expression. Lesson 4: Solving Quadratic Equations I can solve linear and quadratic equations using symbolic reasoning In the last Problem, you explored ways to write quadratic expressions in factored form. In this Problem, you will use the factored form to find solutions to quadratic equations. If you know that the product of two numbers is zero, what can you say about the numbers? How can you solve the equation 0 = x 2 + 8x + 12 by factoring. First, write x 2 + 8x + 12 in factored form to get (x + 2)(x + 6). This expression is the product of two linear factors.

11 When 0 = (x + 2)(x + 6), what must be true about one or both of the linear factors? How can this information help you find the solutions to 0 = (x + 2)(x + 6)? How can this information help you find the x-intercepts of the graph of y = x 2 + 8x + 12? Problem 3.4 A. 1. Write x x + 24 in factored form. 2. How can you use the factored form to solve x x + 24 = 0 for x? 3. Explain how the solutions to 0 = x x + 24 relate to the graph of y = x x B. Solve each equation for x without making a table or graph.

12 C. Solve each equation for x without making a table or graph. Check your answers. D. You can approximate the height h of a pole-vaulter from the ground after t seconds with the equation h = 32t 16t Suppose the pole-vaulter writes the equation 0 = 32t 16t 2. What information is the polevaulter looking for? 2. The pole-vaulter wants to clear a height of 17.5 feet. Will the pole-vaulter clear the desired height? Explain. Practice ACE Questions: # 24-30