1. Complex Numbers and Roots
5-4
Complex Numbers and Roots
SWBAT:
Define and use imaginary and complex numbers.
Solve quadratic equations with complex roots.
Holt Algebra 2

2. Square Roots:
Let’s Review some properties of roots

3. 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.
However, you can find solutions if you define the square root of negative numbers, which is why imaginary numbers were invented. The imaginary unit i is defined as You can use the imaginary unit to write the square root of any negative number.

4. Imaginary Numbers: So far we have used real numbers.
Now we use imaginary numbers: ‘i’
i =
i2 = -1
Ex: = 4i

5. Simplifying Square Roots of Negative Numbers
Express the number in terms of i.

6. Solving a Quadratic Equation with Imaginary Solutions
Solve the equation.
5x = 0
x2 = –36
x = 0
9x = 0

7. 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 part a and an imaginary part b.

8. Equating Two Complex Numbers
Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.
Find the values of x and y that make the equation 4x + 10i = 2 – (4y)i true .

9. Find the values of x and y that make each equation true.
2x – 6i = –8 + (20y)i

10. Homework
1. Page 264 #26-29, #42-49all

11. Warm Up – Writing Equations
Jennifer earns a commission of $1.75 for each magazine subscription she sells and $1.50 for each newspaper subscription she sells. Her goal is to earn a total of $525 in commission in the next two weeks.
Write an equation that is a model for the different numbers of magazine and newspaper subscriptions that can be sold to meet the goal.
If Jennifer sells 100 magazine subscriptions and 200 newspaper subscriptions, will she meet her goal? Explain.

12. 2-9 Operations with Complex Numbers SWBAT
Perform operations with complex numbers.
Holt Algebra 2

13. Adding and subtracting complex numbers is similar to adding and subtracting variable expressions with like terms. Simply combine the real parts, and combine the imaginary parts.
The set of complex numbers has all the properties of the set of real numbers. So you can use the Commutative, Associative, and Distributive Properties to simplify complex number expressions.

14. Adding and Subtracting Complex Numbers
Add or subtract. Write the result in the form a + bi.
a. (4 + 2i) + (–6 – 7i)
Add real parts and imaginary parts.
b. (5 –2i) – (–2 –3i)
c. (1 – 3i) + (–1 + 3i)
d. (–3 + 5i) + (–6i)
e. 2i – (3 + 5i)
f. (4 + 3i) + (4 – 3i)

15. You can multiply complex numbers by using the Distributive Property and treating the imaginary parts as like terms. Simplify by using the fact i2 = –1.

16. Multiplying Complex Numbers
Multiply. Write the result in the form a + bi.
a. –2i(2 – 4i)
b. (3 + 6i)(4 – i)
c. (2 + 9i)(2 – 9i)
d. (–5i)(6i)
e. 2i(3 – 5i)
f. (4 – 4i)(6 – i)

17. The imaginary unit i can be raised to higher powers.
Notice the repeating pattern in each row of the table. The pattern allows you to express any power of i as one of four possible values: i, –1,
–i, or 1.
Helpful Hint

18. Evaluating Powers of i
Simplify –6i14.
–6i14 = –6(i2)7
Rewrite i14 as a power of i2.
= –6(–1)7
= –6(–1) = 6
Simplify.

19. Evaluating Powers of i
Simplify i63.
Simplify i42.

20. Homework
2. Pg 264 # all

21. Warm Up – System of Equations

22. 2-9 Operations with Complex Numbers SWBAT
Perform operations with complex numbers.
Holt Algebra 2

23. Conjugates: What is a conjugate? Factor x2 – 100.
Find the conjugate of the following:
1. (4 + i)
2. (i – 3)
3. (4i)
The complex conjugate of a complex number a + bi is a – bi.
Helpful Hint

24. Dividing Complex Numbers
Simplify.
Multiply by the conjugate.
Distribute.
Use i2 = –1.
Simplify.

25. Simplify.

26. The table shows several complex numbers, where i is the imaginary unit
The table shows several complex numbers, where i is the imaginary unit. Mark all appropriate boxes with an “x” in the table where the product of the two numbers is an imaginary number.
8 – 2i 3 i
3 + 2i 5i -4
conjugate

27. Lesson Quiz: Part I
Perform the indicated operation. Write the result in the form a + bi.
1. (2 + 4i) + (–6 – 4i) –4
2. (5 – i) – (8 – 2i)
–3 + i
3. (2 + 5i)(3 – 2i) 4.
3 + i i
5. Simplify i31. –i

28. Homework
Worksheet