A. Graph the parabola. B. Where are the solutions to the equation, 0= x + 1? C. What does the Fundamental Theorem of Algebra say?

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1 Hart Interactive Honors Algebra 1 Lesson 6 M4+ Opening Exercises 1. Watch the YouTube video Imaginary Numbers Are Real [Part1: Introduction] by Welch Labs ( Fill out the notes below. A. Graph the parabola fx () = x + 1. B. Where are the solutions to the equation, 0= x + 1? C. What does the Fundamental Theorem of Algebra say? D. What is this picture at the right showing? E. Summarize what Gauss said about the name imaginary numbers. S.13 This work is derived from Eureka Math and licensed by Great Minds. 015 Great Minds. eureka-math.org This file derived from ALG II-M1-TE

2 Hart Interactive Honors Algebra 1 Lesson 6 M4+. Use the word bank below to fill in the diagram. Complex Numbers Counting or Natural Numbers Imaginary Numbers Integers Irrational Numbers Rational Numbers Real Numbers Whole Numbers 0, 1,, 3,... and -1, -, -3,... and fractions and decimals that have a pattern or end 0, 1,, 3,... and -1, -, -3,... Cannot be written as a fraction (ratio), like π and 0, 1,, 3,... 1,, 3, What type of number is 1.3? 4? 3? i S.14 This work is derived from Eureka Math and licensed by Great Minds. 015 Great Minds. eureka-math.org This file derived from ALG II-M1-TE

3 Hart Interactive Honors Algebra 1 Lesson 6 M4+ When Will We Get Complex Solutions? 4. The expression under the radical in the quadratic formula, bb 4aaaa, is called the discriminant. Why do you think it is called this? 5. Let s examine quadratic equations and their graphs to determine what the discriminant tells us. Complete the table below. Equation Graph Number and Type of Solution Discriminant A. fx () = x + x 6 B. ff(xx) = xx + 6xx + 9 C. ff(xx) = xx + 4 S.15 This work is derived from Eureka Math and licensed by Great Minds. 015 Great Minds. eureka-math.org This file derived from ALG II-M1-TE

4 Hart Interactive Honors Algebra 1 Lesson 6 M4+ 6. Consider the equation 3xx + xx = 7. Compute the value of the discriminant. Without graphing, what does the value of the discriminant tell us about number and type of solutions to this equation? 7. How does the value of the discriminant for each equation relate the number of solutions you found? How does it relate to the type of solution? 8. Compute the value of the discriminant of the quadratic equation in each part. Use the value of the discriminant to predict the number and type of solutions. Equation Discriminant Number and Type of Solutions A. xx + xx + 1 = 0 B. xx 9 = 0 C. 9xx 4xx 14 = 0 D. 3xx + 4xx + = 0 E. xx = xx + 5 S.16 This work is derived from Eureka Math and licensed by Great Minds. 015 Great Minds. eureka-math.org This file derived from ALG II-M1-TE

5 Hart Interactive Honors Algebra 1 Lesson 6 M4+ Lesson Summary A quadratic equation with real coefficients may have real or complex solutions. Real solutions to polynomial equations correspond to the xx-intercepts of the associated graph, but complex solutions do not. Given a quadratic equation aaxx + bbxx + aa = 0, the discriminant bb 4aaaa indicates whether the equation has two distinct real solutions, one real solution, or two complex solutions. If bb 4aaaa > 0, there are two real solutions to aaxx + bbxx + aa = 0. If bb 4aaaa = 0, there is one real solution to aaxx + bbxx + aa = 0. If bb 4aaaa < 0, there are two complex solutions to aaxx + bbxx + aa = 0. Homework Problem Set 1. Determine the number and type of each solution for the following quadratic equations. A. x 6x+ 8= 0 B. x 8x+ 16= 0 C. 4x + 1= 0. Give a new example of a quadratic equation in standard form that has a. Exactly two distinct real solutions. b. Exactly one distinct real solution. c. Exactly two complex (non-real) solutions. S.17 This work is derived from Eureka Math and licensed by Great Minds. 015 Great Minds. eureka-math.org This file derived from ALG II-M1-TE

6 Hart Interactive Honors Algebra 1 Lesson 6 M4+ 3. Suppose we have a quadratic equation aaxx + bbxx + aa = 0 so that aa + aa = 0. Does the quadratic equation have one solution or two distinct solutions? Are they real or complex? Explain how you know. 4. Write a quadratic equation in standard form such that 5 is its only solution. 5. Is it possible that the quadratic equation aaxx + bbxx + aa = 0 has a positive real solution if aa, bb, and aa are all positive real numbers? A. What are the two solutions to the quadratic equation aaxx + bbxx + aa = 0? B. When will these solutions be positive? 6. Is it possible that the quadratic equation aaxx + bbxx + aa = 0 has a positive real solution if aa, bb, and aa are all negative real numbers? Explain your thinking. S.18 This work is derived from Eureka Math and licensed by Great Minds. 015 Great Minds. eureka-math.org This file derived from ALG II-M1-TE

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