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

3. The Discriminant

4. The Discriminant

5. Example 1

6. Example 1

7. Example 1

8. Example 1

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

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

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

12. 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:

14. Imaginary Numbers

15. Example 2

16. Example 2

17. Try This
Use the quadratic formula to solve:

18. Try This
Use the quadratic formula to solve:

20. Example 3

21. Example 3

22. Try This
Find x and y such that
2x + 3iy = i

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

24. Example 4

25. Example 4

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

27. 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?

28. 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? i

29. Example 5

30. Example 5

31. Try This
Multiply

32. Try This
Multiply

33. 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 i is 2 – 5i and the conjugate of 1 – 3i is 1 + 3i.

34. 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 i is 2 – 5i and the conjugate of 1 – 3i is 1 + 3i.
The conjugate of is denoted

35. 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 i 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.

36. Example 6
Simplify Write your answer in standard form.

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

38. Example 6
Simplify Write your answer in standard form.

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

41. Homework
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24-66 multiples of 3