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Quadratic Equations and Complex Numbers

Quadratic Equations and Complex Numbers. Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers. The Discriminant. The Discriminant. Example 1. Example 1. Example 1. Example 1. Try This.

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Quadratic Equations and Complex Numbers

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  1. Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.

  2. The Discriminant

  3. The Discriminant

  4. Example 1

  5. Example 1

  6. Example 1

  7. Example 1

  8. Try This • Find the discriminant for each equation. Then, determine the number of real solutions.

  9. Try This • Find the discriminant for each equation. Then, determine the number of real solutions. 2 real roots

  10. Try This • Find the discriminant for each equation. Then, determine the number of real solutions. 2 real roots 0 real roots

  11. Imaginary Numbers • If the discriminant is negative, that means when using the quadratic formula, you will have a negative number under a square root. This is what we call an imaginary number and is defined as:

  12. Imaginary Numbers

  13. Example 2

  14. Example 2

  15. Try This • Use the quadratic formula to solve:

  16. Try This • Use the quadratic formula to solve:

  17. Example 3

  18. Example 3

  19. Try This • Find x and y such that 2x + 3iy = -8 + 10i

  20. Try This • Find x and y such that 2x + 3iy = -8 + 10i real part imaginary part

  21. Example 4

  22. Example 4

  23. Additive Inverses • Two complex numbers whose real parts are opposites and whose imaginary parts are opposites are called additive inverses.

  24. Additive Inverses • Two complex numbers whose real parts are opposites and whose imaginary parts are opposites are called additive inverses. • What is the additive inverse of 2 – 12i?

  25. Additive Inverses • Two complex numbers whose real parts are opposites and whose imaginary parts are opposites are called additive inverses. • What is the additive inverse of 2 – 12i? -2 + 12i

  26. Example 5

  27. Example 5

  28. Try This • Multiply

  29. Try This • Multiply

  30. Conjugate of a Complex Number • In order to simplify a fraction containing complex numbers, you often need to use the conjugate of a complex number. For example, the conjugate of 2 + 5i is 2 – 5i and the conjugate of 1 – 3i is 1 + 3i.

  31. Conjugate of a Complex Number • In order to simplify a fraction containing complex numbers, you often need to use the conjugate of a complex number. For example, the conjugate of 2 + 5i is 2 – 5i and the conjugate of 1 – 3i is 1 + 3i. • The conjugate of is denoted .

  32. Conjugate of a Complex Number • In order to simplify a fraction containing complex numbers, you often need to use the conjugate of a complex number. For example, the conjugate of 2 + 5i is 2 – 5i and the conjugate of 1 – 3i is 1 + 3i. • The conjugate of is denoted . • To simplify a quotient with an imaginary number, multiply by 1 using the conjugate of the denominator.

  33. Example 6 • Simplify . Write your answer in standard form.

  34. Example 6 • Simplify . Write your answer in standard form. • Multiply the top and bottom by 2 + 3i.

  35. Example 6 • Simplify . Write your answer in standard form.

  36. Example 6 • Simplify . Write your answer in standard form. • Multiply the top and bottom by 2 – i.

  37. Homework • Page 320 • 24-66 multiples of 3

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