1. Warm-Up Use the quadratic formula to solve each equation. 6 minutes 1) x 2 + 12x + 35 = 02) x 2 + 81 = 18x 3) x 2 + 4x – 9 = 04) 2x 2 = 5x + 9

2. 5.6 Quadratic Equations and Complex Numbers 5.6 Quadratic Equations and Complex Numbers Objectives: Classify and find all roots of a quadratic equation

3. Solutions of a Quadratic Equation If b 2 – 4ac > 0, then the quadratic equation has 2 distinct real solutions. Let ax 2 + bx + c = 0, where a = 0. If b 2 – 4ac = 0, then the quadratic equation has 1 real solutions. If b 2 – 4ac < 0, then the quadratic equation has 0 real solutions. The expression b 2 – 4ac is called the discriminant.

4. Example 1 Find the discriminant for each equation. Then determine the number of real solutions. a) 3x 2 – 6x + 4 = 0 b 2 – 4ac =(-6) 2 – 4(3)(4) =36 – 48 =-12 no real solutions b) 3x 2 – 6x + 3 = 0 b 2 – 4ac =(-6) 2 – 4(3)(3) =36 – 36 =0 one real solution c) 3x 2 – 6x + 2 = 0 b 2 – 4ac =(-6) 2 – 4(3)(2) =36 – 24 =12 two real solutions

5. Practice Identify the number of real solutions: 1) -3x 2 – 6x + 15 = 0

6. Imaginary Numbers The imaginary unit is defined as and i 2 = -1. If r > 0, then the imaginary number is defined as follows:

7. Example 2 Solve 6x 2 – 3x + 1 = 0.

8. Practice Solve -4x 2 + 5x – 3 = 0.

9. Homework p.320 #15-25 odds,29,33,37,43

10. Warm-Up Find the discriminant, and determine the number of real solutions. Then solve. 5 minutes 1) x 2 – 7x = -102) 5x 2 + 4x = -5

11. 5.6 Quadratic Equations and Complex Numbers 5.6 Quadratic Equations and Complex Numbers Objectives: Graph and perform operations on complex numbers

12. Imaginary Numbers A complex number is any number that can be written as a + bi, where a and b are real numbers and a is called the real part and b is called the imaginary part. 3 + 4i real part imaginary part 34i

13. Example 1 Find x and y such that -3x + 4iy = 21 – 16i. Real partsImaginary parts -3x = 21 x = -7 4y = -16 y = -4 x = -7 and y = -4

14. Practice Find x and y such that 2x + 3iy = -8 + 10i.

15. Example 2 Find each sum or difference. a) (-10 – 6i) + (8 – i) = (-10 + 8) = -2 – 7i b) (-9 + 2i) – (3 – 4i) = (-9 – 3) = -12 + 6i + (2i + 4i) + (-6i – i)

16. Example 3 Multiply. (2 – i)(-3 – 4i) = -6- 8i+ 3i+ 4i 2 = -6- 5i+ 4(-1) = -10 – 5i

17. Conjugate of a Complex Number The conjugate of a complex number a + bi is a – bi. The conjugate of a + bi is denoted a + bi.

18. Example 4 multiply by 1, using the conjugate of the denominator = (3 – 2i) (-4 + i) (-4 – i) (-4 - i) = -12 16 - 3i + 4i + 8i+ 2i 2 - 4i- i 2 = -12 16 + 5i+ 2(-1) - (-1) = -14 17 + 5i

19. Practice

20. Warm-Up Perform the indicated operations, and simplify. 5 minutes 1) (-4 + 2i) + (6 – 3i)2) (2 + 5i) – (5 + 3i) 3) (7 + 7i) – (-6 – 2i)4)

21. 5.6.3 Quadratic Equations and Complex Numbers 5.6.3 Quadratic Equations and Complex Numbers Objectives: Graph and perform operations on complex numbers

22. The Complex Plane In the complex plane, the horizontal axis is called the real axis and the vertical axis is called the imaginary axis. -4 -2 2 4 2 4 -4 -2 real axis imaginary axis

23. Homework worksheet 5.6 “B”