Solving Quadratic Equations (Adapted from Core Plus Mathematics, Courses 1 and 2)

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1 Solving Quadratic Equations (Adapted from Core Plus Mathematics, Courses 1 and ) In situations that involve quadratic functions, the interesting questions often require solving equations. For example, When a pumpkin is dropped from a point 50 feet above the ground, it will hit the ground at the time t that satisfies the equation 50-16t = 0. To find points where the main cable of a suspension bridge is 0 feet above the bridge surface, you might need to solve an equation like 0.0x - x = 0. You can always estimate solutions for these equations by scanning tables of (x, y) values or by tracing coordinates of points on function graphs. In some cases, you can get exact solutions by reasoning algebraically--without use of calculator table or graphs. As you work on the problems of this investigation, look for answers to this question: What do we mean by the term solutions to quadratic equations? What strategies can be used to solve quadratic equations algebraically, by factoring, and by using the quadratic fornula? Part I - Solving Quadratics of the Form ax + c = d and ax + bx = 0, Algebraically 1. Some quadratic equations can be solved by use of the fact that for any positive number n, the equation x = n is satisfied by two numbers: n and n. Use this principle and what you know about solving linear equations to solve the following quadratic equations. In each, check your reasoning by substituting the proposed solution values for x in the original equation. a. x = 5 e. -5x + 75 = 60 b. x = 1 f. 5x + 8 = 8

2 c. 5x = 60 g. 5x + 75 = 60 d. 5x + 15 = 60 h. x = -16. Use methods you developed in reasoning to solutions for equations in Problem 1 to answer these questions about flight of a platform diver. a. You may remember that the distance of a free falling object is given by the equation d = 1 gt. If the diver jumps off a 50-foot platform, what rule gives her or his distance fallen d (in feet) as a function of time t (in seconds)? b. Write and solve an equation to find the time required for the diver fall 0 feet. c. What function gives the height h (in feet) as a function of time t (in seconds) after she or he jumps from the platform? d. Write and solve an equation to find the the time when the diver hits the water. 3. If a soccer player kicks the ball from a spot on the ground with initial upward velocity of 4 feet per second, the height of the ball h (in feet at any time t seconds after the kick will be approximated by the quadratic function h(t) = 4t - 16t. a. Explain why finding the time when the ball hits the ground requires that we solve the equation 4t - 16t = 0? b. Check the reasoning in this proposed solution of the equation. i. The expression 4t - 16t is equivalent to 8t(3 - t). Why? ii. The expression 8t(3 - t) will equal 0 when t = 0 and when 3 - t = 0. Why?

3 iii. So, the solutions of the equation 4t - 16t = 0 will be 0 and 1.5. Why? c. Adapt the reasoning in Part a to solve these quadratic equations. i. 0 = x + 4x iv. -x - 5x = 0 ii. 0 = 3x + 10x v. -x - 6x = 0 iii. 0 = x - 4x vi. x + 5x = 6 4. Solving quadratic equations like 3x - 15 = 0 and 3x - 15x = 0 locates x-intercepts on the graphs of the quadratic functions y = 3x - 15 and y = 3x - 15x. a. Using the graphs above explain how the symmetry of these parabolas can be used to relate the location of the minimum (or maximum) point on the graph of a quadratic function to the x-intercepts.

4 b. Use the results of your work in Problem 3 to find coordinates of the maximum or minimum points on the graphs of these quadratic functions. i. y = x + 4x iv. y = -x - 5x ii. y = 3x + 10x v. y = -x - 6x iii. y = x - 4x vi. y = ax + bx

5 5. Use what you know about solving quadratic equations and the graphs of quadratic functions to answer these questions. a. Describe a process that uses rules of algebra to find solutions for any quadratic equation in the form ax = d? i. Complete the table identifying which values of a and d will give you 0, 1, or solutions to the equation ax = d. For solutions to ax = d: Complete the equality Number of Solutions Visual Representation If d and a have opposite signs, then d a 0 If d = 0, then d a 0 If d and a have the same sign, then d a 0

6 b. What algebraic process would you use to find solutions to the equation ax + c = d? i. Complete the table identifying which values of a and d will give you 0, 1, or solutions to the equation ax + c = d. For solutions to ax + c = d: Complete the equality Number of Solutions Visual Representation If (d - c) and a have opposite signs, then d c a 0 If (d - c) = 0, then d c a 0 If (d - c) and a have the same sign, then d c a 0

7 c. What algebraic process would you use to find solutions to the equation ax + bx = 0 (where neither a or b is 0)? i. How many solutions will there be? ii. What are the solutions to the equation ax + bx = 0? iii. Provide a visual representation of such solutions by drawing a graph. d. What algebraic process would you use to find solutions to the equation ax + bx + c = 0 when the expression ax + bx + c can be written as a product of two linear expressions? i. How many solutions will there be? ii. What are the solutions to the equation ax + bx + c = 0? iii. Provide a visual representation of such solutions by drawing a graph. 6. How can you locate the maximum or minimum point on the graph of a quadratic function with rules in the form: a. y = ax b. y = ax + c c. y = ax + bx

8 Part II - Solving Quadratics Using The Quadratic Formula Many problems that require solving quadratic equation involve trinomial expressions like t -16t that are not easily expressed in equivalent factored form. So solving equations like t -16t = 100 (When is a flying pumpkin 100 feet above the ground) is not as easy as solving equations like those in Part I. Fortunately, there is a quadratic formula that shows how to find all solutions of any quadratic equation in the form ax + bx + c = 0. For any such equation, the solutions are b b 4ac b x = + and x = - Or more commonly written as: b 4ac In the next investigation, you will prove that the quadratic formula gives the solutions to any quadratic equation. For now, to use the quadratic formula in any particular case, all you have to do is Be sure that the quadratic expression is set equal to 0 as is prescribed by the formula; Identify the values of a, b, and c; and Substitute those values where they occur in the formula. As you work through the problems in this section, make notes of answers to these questions: What calculations in the quadratic formula give information on the number of solutions of the related quadratic equation? What calculations provide information on the x-intercepts and maximum or minimum point of the graph of the related quadratic function? Use the following procedure for solving quadratics using the quadratic formula. Rewrite the equation in the form ax + bx + c = 0. Identify the values of a, b, and c, and plug them into the quadratic formula. b b 4ac Evaluate and for those values. b b 4ac b b 4ac Evaluate x = + and x = -. Check that the solutions produced by the formula actually satisfy the equation. 7. Use the procedure outlined above to solve the following equations by use of the quadratic formula. In each case, check your work by substituting proposed solutions into the original equation and by sketching a graph of the related quadratic function to show how the solutions appear as points on the graph. a. x - x - 1 = 0

9 b t - 16t = 100 c. x - 7x + 10 = 0 d. x - x - 8 = 0 e. - x - 3x + 10 = 0

10 f. x - 1x + 18 = 0 g. 13-6x + 10 = 0 h. x - 7x + x = 0 8. Study the results of your work in the previous question. a. What part of the quadratic formula calculations shows whether there will be, 1, or 0 real number solutions? b. What specific requirements for that calculation will determine whether there are, 1, or 0 real number solutions?

11 9. The formula for calculating solutions of quadratic equations is a complex set of directions. You can begin to make sense of the formula by connecting it to patterns in the graphs of quadratic functions. Consider the related quadratic functions: f(x) = x - 1x f(x) = x - 1x + 10 f(x) = x - 1x - 14 f(x) = x - 1x + 4 The graphs of these functions are shown in the following diagram. a. Match each function with its graph. b. What are the x-coordinates of the minimum points on each graph? How (if at all) are those x-coordinates related to the x-intercepts of the graphs? c. What are the coordinates of the x-intercepts on the graph of a quadratic function with rule f(x) = ax + bx? What is the x-coordinate of the minimum or maximum point for such a graph? d. How will the x-coordinate of the maximum or minimum point of f(x) = ax + bx + c be related to that of f(x) = ax + bx?

12 10. Now look back at the quadratic formula and think about how the results of Problem 3 help to explain the connections between these parts of the formula and the graph of f(x) = ax + bx + c. Draw and label the graph below according to the following pieces of the quadratic formula. a. b b 4ac b. b c. + b 4ac b d. - b 4ac Summarize the Mathematics A. How can you solve quadratic equations like ax + b = c using algebraic reasoning? B. How can you solve equations like ax + bx = 0 using algebraic reasoning? C. How can you s olve equations like ax + bx + c = 0 when the expression ax + bx + c can be written in equivalent form as the product of two linear expressions?

13 D. In what situations does it make sense to use the quadratic formula to solve an equation? E. What are the key steps in using the quadratic formula to solve a quadratic equation? F. How does the quadratic formula show whether a given quadratic equation will have, 1, or 0 real number solutions? How will this information appear in a graph and in the calculations leading to the solutions? G. How are the locations of the maximum or minimum point and x-intercepts of a quadratic function graph shown by calculations in the quadratic formula?

14 H. Complete the table below to to show how you would choose among solution strategies to solve the different forms of quadratic equations that arise in solving problems: Solution Strategy When to use it Process Estimation using a table or graph Square root method with arithmetic operations Factoring Quadratic Formula

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