1. Warm-Up #9
Find the discriminant and determine the number of real solutions. Then solve.
1) 𝒙 𝟐 −𝟑𝒙=𝟕 2) 𝟒𝒙 𝟐 +𝟗 =𝟏𝟐𝒙

2. Homework
Lesson 5.6_page 320
#48-53 ALL

3. Homework Solution pg 320_#24-29
24. Discriminant=-7, 0 solution; x= −5 2 ± 𝑖 ; 0; x= 5 6 ± 𝑖 ; 2; x =2 and ; 0; x= 1 2 ± 𝑖 ; 0; x=4±𝑖 ; 2; x= 3 2 ± 37 2

4. 5.6 Complex Numbers

5. Property of the square root of negative numbers
If r is a positive real number, then
Examples:

6. −1 =𝑖 𝑖 5 =𝑖 𝑖 2 =−1 6=−1 𝑖 3 =−𝑖 𝑖 7 =−𝑖 𝑖 4 =1 𝑖 8 =1
The “Imaginary Rule”
−1 =𝑖
𝑖 2 =−1
𝑖 3 =−𝑖
𝑖 4 =1
𝑖 5 =𝑖
6=−1
𝑖 7 =−𝑖
𝑖 8 =1

7. *For larger exponents, divide the exponent by 4, then use the remainder as your exponent instead.
Example:

8. Examples
( 𝑖 5) 2
𝑖 4 23

9. Complex Numbers Example: 5+4i, 4-3i
A complex number has a real part & an imaginary part.
Standard form is:
Real part
Imaginary part
Example: 5+4i, 4-3i

10. Identify Real Part and Imaginary Part
Complex Numbers
Real Part
Imaginary Part
6 – 6i
7+ 3i
-13 – 5i

11. Adding and Subtracting: add or subtract the real parts, then add or subtract the imaginary parts
1. 𝟑 𝟖 + 𝟐 𝟑 𝒊 −( 𝟏 𝟒 − 𝟏 𝟑 𝒊)
2. (-2+3i)+(-2—4i)

12. Adding and Subtracting: add or subtract the real parts, then add or subtract the imaginary parts
1. 𝟏 𝟐 + 𝟑 𝟕 𝒊 −( 𝟑 𝟒 − 𝟏 𝟑 𝒊)
2. (-9+i)+(-12—6i)

13. Multiplying: Treat the i’s like variables, then apply the “imaginary rule”
1. (-5+3i)(2—3i)
2. (-5—i)(-2+2i)

14. Multiplying: Treat the i’s like variables, then apply the “imaginary rule”
1. (-7+2i)(6—8i)
2. (1—2i)(-5+9i)

15. Expanding: How do you do this?
(−𝟏−𝟑𝒙)(𝟐−𝒙) (𝟐−𝒙)(𝟐+𝒙)
(−𝟏−𝟑𝒊)(𝟐−𝒊) (𝟐−𝒊)(𝟐+𝒊)

16. You Try!
(𝟐+𝒊)(−𝟑+𝒊) (−𝟑−𝒊)(−𝟑+𝒊)
(−𝟒+𝒊)(𝟏−𝒊) (𝟏+𝒊)(𝟏−𝒊)

17. Adding and Subtracting (add or subtract the real parts, then add or subtract the imaginary parts)
Ex:
Ex:
Ex:

18. Multiplying Treat the i’s like variables, then apply the “imaginary rule”
Ex:
Ex: