1. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 Chapter 3 Quadratic Functions and Equations

2. 2 Copyright © 2014, 2010, 2006 Pearson Education, Inc. Complex Numbers ♦ Perform arithmetic operations on complex numbers ♦ Solve quadratic equations having complex solutions 3.3

3. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 3 Properties of the Imaginary Unit i Defining the number i allows us to say that the solutions to the equation x 2 + 1 = 0 are i and –i.

4. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 4 Complex Numbers A complex number can be written in standard form as a + bi where a and b are real numbers. The real part is a and the imaginary part is b. Every real number a is also a complex number because it can be written as a + 0i.

5. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 5 Imaginary Numbers A complex number a + bi with b ≠ 0 is an imaginary number. A complex number a + bi with a = 0 and b ≠ 0 is sometimes called a pure imaginary number. Examples of pure imaginary numbers include 3i and –i.

6. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 6 The Expression If a > 0, then

7. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 7 Example: Simplifying expressions Simplify each expression. Solution

8. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 8 Example: Performing complex arithmetic Write each expression in standard form. Support your results using a calculator. a) (  3 + 4i) + (5  i)b) (  7i)  (6  5i) c) (  3 + 2i) 2 d) Solution a) (  3 + 4i) + (5  i) =  3 + 5 + 4i  i = 2 + 3i b) (  7i)  (6  5i) =  6  7i + 5i =  6  2i

9. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 9 Example: Performing complex arithmetic c) (  3 + 2i) 2 = (  3 + 2i)(  3 + 2i) = 9 – 6i – 6i + 4i 2 = 9  12i + 4(  1) = 5  12i d)

10. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 10 Quadratic Equations with Complex Solutions We can use the quadratic formula to solve quadratic equations if the discriminant is negative. There are no real solutions, and the graph does not intersect the x-axis. The solutions can be expressed as imaginary numbers.

11. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 11 Example: Solving a quadratic Solve the quadratic equation x 2 + 3x + 5 = 0. Support your answer graphically. Solution a = 1, b = 3, c = 5

12. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 12 Example: Solving a quadratic Solution continued The graph does not intersect the x-axis, so no real solutions, but two complex solutions that are imaginary.

13. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 13 Example: Solving a quadratic Solve the quadratic equation Support your answer graphically. Solution Rewrite the equation: a = 1/2, b = –5, c = 17

14. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 14 Example: Solving a quadratic Solution continued The graphs do not intersect, so no real solutions, but two complex solutions that are imaginary.

15. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 15 Example: Solving a quadratic Solve the quadratic equation –2x 2 = 3. Support your answer graphically. Solution Apply the square root property.

16. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 16 Example: Solving a quadratic Solution continued The graphs do not intersect, so no real solutions, but two complex solutions that are imaginary.