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Chapter 2

Chapter 2 . 2-5 complex numbers and roots. Objectives . Define and use imaginary and complex numbers Solve quadratic equations with complex roots. Imaginary numbers. Can we solve the following equation? Does it have real zeros?

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Chapter 2

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  1. Chapter 2 2-5 complex numbers and roots

  2. Objectives • Define and use imaginary and complex numbers • Solve quadratic equations with complex roots

  3. Imaginary numbers • Can we solve the following equation? • Does it have real zeros? • You can see in the graph of f(x) = x2 + 1 below that f has no real zeros. If you solve the corresponding equation 0 = x2 + 1, you find that x = ,,which has no real solutions.

  4. Imaginary numbers • However, you can find solutions if you define the square root of negative numbers, which is why imaginary numbers were invented. • What is an imaginary number or imaginary unit? • The imaginary uniti is defined as • You can use the imaginary unit to write the square root of any negative number.

  5. Imaginary numbers

  6. Example #1 • Express the number in terms of i.

  7. Example #2 Express the number in terms of i

  8. Example #3 • Express the number in terms of i.

  9. Student guided practice • Teacher: work on problem #2 from worksheet • Students: work on worksheet

  10. Solve Equations with imaginary numbers • Example #4 • Solve the equation

  11. Example #5 • Solve the equation. • x2 = –36

  12. Example #6 • Solve the equations. • x2 + 48 = 0 • 9x2 + 25 = 0

  13. Student guided practice • work on problems 4-7 from practice b worksheet

  14. What is a complex number? • A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i= . The set of real numbers is a subset of the set of complex numbers C. • Every complex number has a real parta and an imaginary partb.

  15. Example #7 • Find the values of x and y that make the equation 4x + 10i = 2 – (4y)itrue . • Real parts • 4x + 10i = 2 – (4y) • Imaginary parts

  16. Example #8 • Find the values of x and y that make each equation true. • 2x – 6i = –8 + (20y)i • Real parts • 2x– 6i = –8 + (20y)i • Imaginary parts

  17. Finding complex zeros • Find the zeros of the function. • f(x) = x2 + 10x + 26

  18. Example #9 • Find the zeros of the function • f(x) = x2 + 4x + 13

  19. Conjugates • What are conjugates? • The solutions and are related. These solutions are a complex conjugate pair. Their real parts are equal and their imaginary parts are opposites. The complex conjugate of any complex number a + bi is the complex number a – bi. • If a quadratic equation with real coefficients has nonreal roots, those roots are complex conjugates.

  20. Example #10 • Find each complex conjugate. • A. 8 + 5i • B. 6i

  21. Student guided practice Work on practice b worksheet

  22. Homework • Do even problems from 22-35 page 97

  23. Closure • Today we learn about complex roots and how we can use the square root to solve equations. We also learn about conjugates. Next class we are going to learn about the quadratic formula

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