1. Lesson 5–5/5–6 Objectives
Be able to define and use imaginary and complex numbers
Be able to solve quadratic equations with complex roots
Be able to solve quadratic equations using the Quadratic Formula

3. Example 1: Simplifying Square Roots of Negative Numbers
Express the number in terms of i.

4. Example 1A: Simplifying Square Roots of Negative Numbers
Express the number in terms of i.
Factor out –1.
Product Property.
Simplify.
Multiply.
Express in terms of i.

5. Example 1B: Simplifying Square Roots of Negative Numbers
Express the number in terms of i.
Factor out –1.
Product Property.
Simplify.
Express in terms of i.

6. Express the number in terms of i.
Check It Out! Example 1a
Express the number in terms of i.
Factor out –1.
Product Property.
Product Property.
Simplify.
Express in terms of i.

7. Express the number in terms of i.
Check It Out! Example 1b
Express the number in terms of i.
Factor out –1.
Product Property.
Simplify.
Multiply.
Express in terms of i.

8. Express the number in terms of i.
Check It Out! Example 1c
Express the number in terms of i.
Factor out –1.
Product Property.
Simplify.
Multiply.
Express in terms of i.

9. Example 2: Solving a Quadratic Equation with Imaginary Solutions
Solve the equation.
x2 = –144
5x = 0
x2 = –36

10. Example 2A: Solving a Quadratic Equation with Imaginary Solutions
Solve the equation.
Take square roots.
Express in terms of i.
Check
x2 = –144
–144
(12i)2
144i 2
144(–1) 
x2 = –144
–144
144(–1)
144i 2
(–12i)2 

11. Example 2B: Solving a Quadratic Equation with Imaginary Solutions
Solve the equation.
5x = 0
Add –90 to both sides.
Divide both sides by 5.
Take square roots.
Express in terms of i.
5x = 0
5(18)i 2 +90
90(–1) +90 
Check

12. Check It Out! Example 2a Solve the equation. x2 = –36 Check x2 = –36
Take square roots.
Express in terms of i.
Check
x2 = –36
–36
(6i)2
36i 2
36(–1) 
(–6i)2

13. A complex number is a number that can be written in the form a + bi
A complex number is a number that can be written in the form a + bi. 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.

14. Example 4: Finding Complex Zeros of Quadratic Functions
Find the zeros of the function.
f(x) = x2 + 10x + 26
g(x) = x2 + 4x + 12
f(x) = x2 + 4x + 13
g(x) = x2 – 8x + 18

15. Example 4A: Finding Complex Zeros of Quadratic Functions
Find the zeros of the function.
f(x) = x2 + 10x + 26
x2 + 10x + 26 = 0
Set equal to 0.
x2 + 10x = –26 +
Rewrite.
x2 + 10x + 25 = –
Add to both sides.
(x + 5)2 = –1
Factor.
Take square roots.
Simplify.

16. Example 4B: Finding Complex Zeros of Quadratic Functions
Find the zeros of the function.
g(x) = x2 + 4x + 12
x2 + 4x + 12 = 0
Set equal to 0.
x2 + 4x = –12 +
Rewrite.
x2 + 4x + 4 = –12 + 4
Add to both sides.
(x + 2)2 = –8
Factor.
Take square roots.
Simplify.

17. Find the zeros of the function.
Check It Out! Example 4a
Find the zeros of the function.
f(x) = x2 + 4x + 13
x2 + 4x + 13 = 0
Set equal to 0.
x2 + 4x = –13 +
Rewrite.
x2 + 4x + 4 = –13 + 4
Add to both sides.
(x + 2)2 = –9
Factor.
Take square roots.
x = –2 ± 3i
Simplify.

18. Find the zeros of the function.
Check It Out! Example 4b
Find the zeros of the function.
g(x) = x2 – 8x + 18
x2 – 8x + 18 = 0
Set equal to 0.
x2 – 8x = –18 +
Rewrite.
x2 – 8x + 16 = –
Add to both sides.
Factor.
Take square roots.
Simplify.

19. You have learned several methods for solving quadratic equations: graphing/tables, factoring, using square roots, and completing the square. Another method is to use the Quadratic Formula, which allows you to solve ANY quadratic equation in standard form.

20. You can use the Quadratic Formula to solve ANY quadratic equation that is written in standard form, including equations with real solutions or complex solutions.

21. Example 1: Quadratic Functions with Real Zeros
Find the zeros of each function using the Quadratic Formula.
f(x) = 2x2 – 16x + 27
f(x) = x2 + 3x – 7
f(x) = x2 – 8x + 10

22. Example 1: Quadratic Functions with Real Zeros
Find the zeros of f(x)= 2x2 – 16x + 27 using the Quadratic Formula.
2x2 – 16x + 27 = 0
Set f(x) = 0.
Write the Quadratic Formula.
Substitute 2 for a, –16 for b, and 27 for c.
Simplify.
Write in simplest form.

23. Example 1 Continued
Check Solve by completing the square. 

24. Check It Out! Example 1a
Find the zeros of f(x) = x2 + 3x – 7 using the Quadratic Formula.
x2 + 3x – 7 = 0
Set f(x) = 0.
Write the Quadratic Formula.
Substitute 1 for a, 3 for b, and –7 for c.
Simplify.
Write in simplest form.

25. Check It Out! Example 1a Continued
Check Solve by completing the square.
x2 + 3x – 7 = 0
x2 + 3x = 7 

26. Check It Out! Example 1b
Find the zeros of f(x)= x2 – 8x + 10 using the Quadratic Formula.
x2 – 8x + 10 = 0
Set f(x) = 0.
Write the Quadratic Formula.
Substitute 1 for a, –8 for b, and 10 for c.
Simplify.
Write in simplest form.

27. Check It Out! Example 1b Continued
Check Solve by completing the square.
x2 – 8x + 10 = 0
x2 – 8x = –10
x2 – 8x + 16 = –
(x + 4)2 = 6 

28. Example 2: Quadratic Functions with Complex Zeros
Find the zeros of each function using the Quadratic Formula.
f(x)= 4x2 + 3x + 2
f(x)= 3x2 – x + 8

29. Example 2: Quadratic Functions with Complex Zeros
Find the zeros of f(x) = 4x2 + 3x + 2 using the Quadratic Formula.
f(x)= 4x2 + 3x + 2
Set f(x) = 0.
Write the Quadratic Formula.
Substitute 4 for a, 3 for b, and 2 for c.
Simplify.
Write in terms of i.

30. Check It Out! Example 2
Find the zeros of g(x) = 3x2 – x + 8 using the Quadratic Formula.
Set f(x) = 0
Write the Quadratic Formula.
Substitute 3 for a, –1 for b, and 8 for c.
Simplify.
Write in terms of i.

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