Solving Quadratic Equations

Transcription

1 Student Probe Probe 1 Solve. Probe 2 Solve. Probe 3 Solve. Solving Quadratic Equations (Refer to Part 1 of the lesson.) (Refer to Part 2 of the lesson.) (Refer to Part 3 of the lesson.) Lesson Description In this three part lesson, students will learn to solve quadratic equations by extracting square roots, by factoring, and by using the quadratic formula. Each part of the lesson can be taught independently, based upon the needs of the student. Rationale Quadratic equations are the simplest form of polynomial equations to solve and are usually encountered once students are proficient in solving linear equations in one variable. Solving quadratic equations by extracting square roots is efficient, but is dependent upon the form of the equation. Solutions by factoring are efficient and the factors determine the zeros of the function. However, many quadratic equations are difficult or impossible to factor. The quadratic formula will solve all quadratic equations provided the equation is in the form Preparation None. At a Glance What: Solving quadratic equations Common Core State Standard: CC.9-12.A.REI.4 Solve equations and inequalities in one variable. Solve quadratic equations in one variable. Matched Arkansas Standard: AR.9-12.QEF.AII.3.3 (QEF.3.AII.3) Analyze and solve quadratic equations with and without appropriate technology by: - - factoring, - - graphing, - - extracting the square root, - - completing the square, - - using the quadratic formula Mathematical Practices: Make sense of problems and persevere in solving them. Look for and make use of structure. Who: Students who cannot solve quadratic equations Grade Level: Algebra 1 Prerequisite Vocabulary: square root, absolute value Prerequisite Skills: solve linear equations in one variable, factor quadratic trinomials Delivery Format: individual, small group, whole group Lesson Length: 30 minutes for each part Materials, Resources, Technology: graphing calculator (optional) Student Worksheets: none

2 Lesson The Expect students to say or do If students do not, then the Part 1: Solve Quadratic Equations by Extracting Square Roots 1. We are going to solve quadratic equations. In some ways solving quadratic equations is like solving linear equations. 2. When we solve equations we must undo operations. What undoes squaring a number? 3. In some ways solving quadratic equations is different than solving linear equations. What does the graph of a quadratic function look like? 4. Since the graph of a quadratic function is a parabola, how many solutions can a quadratic equation have? 5. Solve. 6. Solve. Taking the square root of the number. It is a parabola. Refer to Solving Equations. Refer to Relating Quadratic Functions and Graphs. 0, 1, or 2 Sketch a graph of a parabola showing the three possibilities. What number times itself equals 4? Is 2 the only number times itself that is equal to 4? What about? 7. Repeat Steps 5-6 with additional equations in the form, as necessary.

3 The Expect students to say or do If students do not, then the 8. Solve. 9. Repeat Step 8 with additional equations in the form, as necessary. Part 2: Solve Quadratic Equations by Factoring 10. We are going to solve quadratic equations. In some ways solving quadratic equations is like solving linear equations. 11. When we solve equations we must undo operations. What undoes squaring a number? 12. In some ways solving quadratic equations is different than solving linear equations. What does the graph of a quadratic function look like? 13. Since the graph of a quadratic function is a parabola, how many solutions can a quadratic equation have? Taking the square root of the number. It is a parabola. Refer to Solving Equations. Refer to Relating Quadratic Functions and Graphs. 0, 1, or 2 Sketch a graph of a parabola showing the three possibilities.

4 The Expect students to say or do If students do not, then the 14. What do we know if the One or both of the numbers If, what is x? product of two numbers is 0? This is called the Zero Product Theorem. It is stated as: must be 0. How do you know? If, then We use this theorem when we solve quadratic equations by factoring. 15. Solve. We have the product of two quantities (or numbers) equal to 0. What can we say? How do you know? 16. Now we have easy equations to solve: and Solve them. 17. Solve. Can we factor the quadratic expression 18. Now we can use the Zero Product Theorem to solve it. Either or The Zero Product Theorem tells us. Yes. Refer to Step 14. Refer to Solving Equations. Refer to Factoring Quadratic Trinomials. Model for students. 19. Repeat Steps with additional equations, as necessary. Part 3: Solve Quadratic Equations using the Quadratic Formula 20. We are going to solve quadratic equations. In some ways solving quadratic equations is like solving linear equations.

5 The Expect students to say or do If students do not, then the 21. When we solve equations we must undo operations. What undoes squaring a number? Taking the square root of the number. Refer to Solving Equations. 22. In some ways solving quadratic equations is different than solving linear equations. What does the graph of a quadratic function look like? 23. Since the graph of a quadratic function is a parabola, how many solutions can a quadratic equation have? 24. We are going to use a method of solving quadratic equations that always works the Quadratic Formula. In order for us to use the formula, the equation must be written in the form. The formula is. It is a parabola. Refer to Relating Quadratic Functions and Graphs. 0, 1, or 2 Sketch a graph of a parabola showing the three possibilities. 25. Let s solve using the formula. What is a? What is b? What is c? 26. Substitute the values into the formula. For this equation, Students may forget that the implied coefficient of and is 1. Monitor students. Model for students, if necessary.

6 The Expect students to say or do If students do not, then the 27. Now we just need to simplify this expression. What are the values for x? Monitor students. Students may need to review order of operations. See Teacher Note 4 for questions concerning. 28. Solve using the quadratic formula. Monitor students. 29. Repeat Steps with additional equations, as necessary.

7 Teacher Notes: 1. Quadratic functions may have 0, 1, or 2 zeros as shown. No Solutions One Solution Two Solutions 2. Most students will solve simple quadratic equations such as by inspection. 3. A common misconception is. By definition, the square root of a number is positive. Thus, and. The two solutions arise from the absolute value of x, rather than from the. 4. The symbol is read plus or minus. It is a shorthand notation indicating positive 2 and negative 2 (or some other quantity. If students find it confusing, they may write instead. 5. Make sure that students understand the quadratic equation must be equal to zero to solve using both factoring or the quadratic formula. 6. To avoid incorrect application of the quadratic formula, have students explicitly the values of a, b, and c. 7. Once students have correctly substituted values into the quadratic formula, you may wish to allow them to use a calculator to obtain the solutions. 8. The quadratic formula may be derived by solving the general quadratic by completing the square. Many texts provide the derivation. 9. Students should be encouraged to check their answers, either by substitution or by graphing. 10. Every effort should be made for students to have the opportunity to connect the solutions to quadratic equations to the graphs of the corresponding quadratic function. Variations 1. Encourage students to use the graphing calculator to determine the number of solutions and the type of solutions (rational, irrational, etc.) for quadratic equations. 2. Once students are confident in their ability to solve quadratic equations, provide them with a variety of equations that may or may not be solved by extracting square roots or factoring. Have students determine the best method for solving each equation.

8 Formative Assessment Part 1 Solve. Part 2 Solve by factoring. Part 3 Solve using the quadratic formula. References Mathematics Intervention at the Secondary Prevention Level of a Multi- Tier Prevention System: Six Key Principles. (n.d.). Retrieved May 13, 2011, from rtinetwork: Paulsen, K., & the IRIS Center. (n.d.). Algebra (part 2): Applying learning strategies to intermediate algebra. Retrieved on October 18, 2011 from pdfhttp://iris.peabody.vanderbilt.edu/case_studies/ICS- 010.pdf