1. Holt McDougal Algebra 2 2-6 The Quadratic Formula Warm Up Write each function in standard form. Evaluate b 2 – 4ac for the given values of the valuables. 1. f(x) = (x – 4) 2 + 3 2. g(x) = 2(x + 6) 2 – 11 f(x) = x 2 – 8x + 19 g(x) = 2x 2 + 24x + 61 4. a = 1, b = 3, c = –3 3. a = 2, b = 7, c = 5 9 21

2. Holt McDougal Algebra 2 2-6 The Quadratic Formula Solve quadratic equations using the Quadratic Formula. Classify roots using the discriminant. Objectives

3. Holt McDougal Algebra 2 2-6 The Quadratic Formula

4. Holt McDougal Algebra 2 2-6 The Quadratic Formula The symmetry of a quadratic function is evident in the last step,. These two zeros are the same distance,, away from the axis of symmetry,,with one zero on either side of the vertex.

5. Holt McDougal Algebra 2 2-6 The Quadratic Formula 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.

6. Holt McDougal Algebra 2 2-6 The Quadratic Formula Find the zeros of f(x)= 2x 2 – 16x + 27 using the Quadratic Formula. Example 1: Quadratic Functions with Real Zeros Set f(x) = 0. Write the Quadratic Formula. Substitute 2 for a, –16 for b, and 27 for c. Simplify. Write in simplest form. 2x 2 – 16x + 27 = 0

7. Holt McDougal Algebra 2 2-6 The Quadratic Formula Find the zeros of f(x) = 4x 2 + 3x + 2 using the Quadratic Formula. Example 2: Quadratic Functions with Complex Zeros 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. f(x)= 4x 2 + 3x + 2

8. Holt McDougal Algebra 2 2-6 The Quadratic Formula The discriminant is part of the Quadratic Formula that you can use to determine the number of real roots of a quadratic equation.

9. Holt McDougal Algebra 2 2-6 The Quadratic Formula

10. Holt McDougal Algebra 2 2-6 The Quadratic Formula Find the type and number of solutions for the equation. Example 3A: Analyzing Quadratic Equations by Using the Discriminant x 2 + 36 = 12x x 2 – 12x + 36 = 0 b 2 – 4ac (–12) 2 – 4(1)(36) 144 – 144 = 0 b 2 – 4ac = 0 The equation has one distinct real solution.

11. Holt McDougal Algebra 2 2-6 The Quadratic Formula Find the type and number of solutions for the equation. Example 3B: Analyzing Quadratic Equations by Using the Discriminant x 2 + 40 = 12x x 2 – 12x + 40 = 0 b 2 – 4ac (–12) 2 – 4(1)(40) 144 – 160 = –16 b 2 –4ac < 0 The equation has two distinct nonreal complex solutions.

12. Holt McDougal Algebra 2 2-6 The Quadratic Formula Find the type and number of solutions for the equation. Example 3C: Analyzing Quadratic Equations by Using the Discriminant x 2 + 30 = 12x x 2 – 12x + 30 = 0 b 2 – 4ac (–12) 2 – 4(1)(30) 144 – 120 = 24 b 2 – 4ac > 0 The equation has two distinct real solutions.

13. Holt McDougal Algebra 2 2-6 The Quadratic Formula The graph shows related functions. Notice that the number of real solutions for the equation can be changed by changing the value of the constant c.

14. Holt McDougal Algebra 2 2-6 The Quadratic Formula An athlete on a track team throws a shot put. The height y of the shot put in feet t seconds after it is thrown is modeled by y = –16t 2 + 24.6t + 6.5. The horizontal distance x in between the athlete and the shot put is modeled by x = 29.3t. To the nearest foot, how far does the shot put land from the athlete? Example 4: Sports Application

15. Holt McDougal Algebra 2 2-6 The Quadratic Formula Step 1 Use the first equation to determine how long it will take the shot put to hit the ground. Set the height of the shot put equal to 0 feet, and the use the quadratic formula to solve for t. y = –16t 2 + 24.6t + 6.5 Example 4 Continued Set y equal to 0. 0 = –16t 2 + 24.6t + 6.5 Use the Quadratic Formula. Substitute –16 for a, 24.6 for b, and 6.5 for c.

16. Holt McDougal Algebra 2 2-6 The Quadratic Formula Example 4 Continued Simplify. The time cannot be negative, so the shot put hits the ground about 1.8 seconds after it is released.

17. Holt McDougal Algebra 2 2-6 The Quadratic Formula Step 2 Find the horizontal distance that the shot put will have traveled in this time. x = 29.3t Example 4 Continued Substitute 1.77 for t. Simplify. x ≈ 29.3(1.77) x ≈ 51.86 x ≈ 52 The shot put will have traveled a horizontal distance of about 52 feet.

18. Holt McDougal Algebra 2 2-6 The Quadratic Formula Check Use substitution to check that the shot put hits the ground after about 1.77 seconds. Example 4 Continued The height is approximately equal to 0 when t = 1.77. y = –16t 2 + 24.6t + 6.5 y ≈ –16(1.77) 2 + 24.6(1.77) + 6.5 y ≈ –50.13 + 43.54 + 6.5 y ≈ –0.09

19. Holt McDougal Algebra 2 2-6 The Quadratic Formula Properties of Solving Quadratic Equations

20. Holt McDougal Algebra 2 2-6 The Quadratic Formula Properties of Solving Quadratic Equations

21. Holt McDougal Algebra 2 2-6 The Quadratic Formula Group Work Group work problems

22. Holt McDougal Algebra 2 2-6 The Quadratic Formula Homework Section 2-6 in the workbook Workbook page 81: 1 - 10