Quadratic Equations; Equations And Inequalities; AII

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Mathematics Enhanced Scope and Sequence – Algebra IIQuadratic EquationsReporting Category Equations and InequalitiesTopicSolving quadratic equations over the set of complex numbersPrimary SOLAII.4b The student will solve, algebraically and graphically, quadraticequations over the set of complex numbers. Graphing calculators willbe used for solving and for confirming the algebraic solutions.MaterialsGraphing calculatorsThree attached handoutsVocabularyquadratic, complex number, imaginary number, factor, zero, quadratic formula, x-intercepts(earlier grades)solving over the set of complex numbers, nature of solution, conjugate, discriminant (AII.4b)Student/Teacher Actions (what students and teachers should be doing to facilitate learning)1. Discuss with the class various ways students have solved quadratic equations in the past.The methods should include factoring, quadratic formula, and graphing. It might alsoinclude completing the square.2. Display the examples found on the attached Different Methods of Solving Quadraticshandout. Model solving each problem, with students reproducing the solution on paper.Discuss the methods used.3. Distribute copies of the attached Solving Quadratics handout. Have students work in pairsto complete it, with Student A solving the odd-number problems and Student B solving theeven-number ones. Then, have partners swap problems and check solutions by substitutingthe solutions back into the original equations. Circulate and support students as they work.24. Once students have developed some confidence,D b 4acNature of the solutionshave them discuss their observations regardingD 0the discriminant and the number and types ofD 0D 0solutions. Have them complete the table at right,D aperfectsquarein which the discriminant D.AssessmentQuestionso How are the graph of a quadratic function and the discriminant of thecorresponding quadratic equation related?2o For quadratic equations in the form ax bx c 0, what effect do changes in eachof the constants have on the number and nature of the solutions?Journal/Writing Promptso Jessica does not know how to solve a quadratic equation, and you must teach her.Write the steps you will tell her to use.Virginia Department of Education 20111

Mathematics Enhanced Scope and Sequence – Algebra IIoList all the various ways to solve a quadratic equation, describe which one you thinkis best, and explain why.OtherooOnce students have had experience solving quadratics with complex solutions, askthem to make observations. They should observe that if a complex number in theform a bi is a solution, then a bi is also a solution—that is, such complex numbersolutions appear in conjugate pairs. Have students explain why.Have students complete the attached Creating Quadratic Equations handout, orgive them your own set of solutions to quadratic equations, and have them creatematching equations.Strategies for DifferentiationBefore students solve quadratic equations, have them identify a, b, and c and substitutethese constants into a part of the quadratic formula. Then, have them substitute them intothe entire formula.Have students sing a song to help them remember the quadratic formula.Review simplifying radicals, as needed.Virginia Department of Education 20112

Mathematics Enhanced Scope and Sequence – Algebra IIDifferent Methods of Solving Quadratics1. Use factoring to solve 2x 2x 1 0.2. Use the quadratic formula to solve 2x 23. Use graphing to solve 2x 2x 1 0.x 1 0.4. Graph yx21 , and discuss ways to solve x 2 1 0 . Use all three methods to solve.5. Graph yx21 , and discuss ways to solve x 26. Use the quadratic formula to solve x 21 0 . Use the quadratic formula to solve.x 1 0.7. Determine the discriminant of 3x 2then solve.5x 2 0 , determine the best method of solving, and8. Determine the discriminant of 3x 2solve.x 2 0 , determine the best method of solving, and thenVirginia Department of Education 20113

Mathematics Enhanced Scope and Sequence – Algebra IISolving QuadraticsFor each of the following quadratics, first determine the discriminant, and then determine the bestmethod of solving and solve. Show all steps.1. 2x2 4x 15 02. 5x2 2x 83. 6x2 x 24 04. 15x2 2x 1 05. 9x2 3x 4 06. 3x2 2x 4 07. 3x2 2x 1 08. 2x2 3x 89. 3x2 4x 210. 2x2 3x 5 011. 3x(x 1) x 512. 2x2 8 x13. 7x 13 x214. x2 3x 5 015. x2 4 2xVirginia Department of Education 20114

Mathematics Enhanced Scope and Sequence – Algebra IICreating Quadratic EquationsConstruct a quadratic equation for each of the following solutions.1. 2 and 55 and2.13.234. 3i and5and1233i5. 2 5i and 2 5i6.12i 31and22i 32Virginia Department of Education 20115

Quadratic Equations Reporting Category Equations and Inequalities Topic Solving quadratic equations over the set of complex numbers Primary SOL AII.4b The student will solve, algebraically and graphically, quadratic equations over the set of complex numbers. Graphing calculators will be used for solving and for confirming the algebraic solutions.

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