1. 9-9 The Discriminant Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

2. 9-9 The Discriminant Warm Up Use the Quadratic Formula to solve each equation. 1. x 2 – 5x – 6 = 0 2. 2x 2 + 2x – 24 = 0 3. x 2 + 10x + 25 = 0 0, –5 3, –4 1, –6

3. 9-9 The Discriminant California Standards 22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x- axis in zero, one, or two points. 23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.

4. 9-9 The Discriminant discriminant Vocabulary

5. 9-9 The Discriminant If the quadratic equation is in standard form, its discriminant is b 2 – 4ac. Notice that this is the expression under the square root in the Quadratic Formula. Recall that quadratic equations can have two, one, or no real solutions. You can determine the number of solutions of a quadratic equation by evaluating the discriminant.

6. 9-9 The Discriminant

7. 9-9 The Discriminant

8. 9-9 The Discriminant Additional Example 1A: Using the Discriminant Find the number of solutions of 3x 2 – 2x + 2 = 0 by using the discriminant. a = 3, b = –2, c = 2 b 2 – 4ac =(–2) 2 – 4(3)(2) = 4 – 24 = –20 Identify the values of a, b, and c. Substitute 3, –2, and 2 for a, b, and c. Simplify. b 2 – 4ac is negative. There are no real solutions.

9. 9-9 The Discriminant Additional Example 1B: Using the Discriminant Find the number of solutions of 2x 2 + 11x + 12 = 0 by using the discriminant. a = 2, b = 11, c = 12 b 2 – 4ac =11 2 – 4(2)(12) = 121 – 96 = 25 Identify the values of a, b, and c. Substitute 2, 11, and 12 for a, b, and c. Simplify. b 2 – 4ac is positive. There are two solutions.

10. 9-9 The Discriminant Additional Example 1C: Using the Discriminant Find the number of solutions of x 2 + 8x + 16 = 0 by using the discriminant. a = 1, b = 8, c = 16 b 2 – 4ac =8 2 – 4(1)(16) = 64 – 64 = 0 Identify the values of a, b, and c. Substitute 1, 8, and 16 for a, b, and c. Simplify. b 2 – 4ac is zero. There is one solution.

11. 9-9 The Discriminant Check It Out! Example 1a Find the number of solutions of 2x 2 – 2x + 3 = 0 using the discriminant. a = 2, b = –2, c = 3 b 2 – 4ac =(–2) 2 – 4(2)(3) = 4 – 24 = –20 Identify the values of a, b, and c. Substitute 2, –2, and 3 for a, b, and c. Simplify. b 2 – 4ac is negative. There are no real solutions.

12. 9-9 The Discriminant Check It Out! Example 1b Find the number of solutions of x 2 + 4x + 4 = 0 using the discriminant. a = 1, b = 4, c = 4 b 2 – 4ac =4 2 – 4(1)(4) = 16 – 16 = 0 Identify the values of a, b, and c. Substitute 1, 4, and 4 for a, b, and c. Simplify. b 2 – 4ac is zero. There is one real solution.

13. 9-9 The Discriminant Recall that the solutions to a quadratic are the same as the x-intercepts of the related function. The discriminant can be used to find the number of x-intercepts.

14. 9-9 The Discriminant Additional Example 2A: Using the Discriminant to Find the number of x-Intercepts Find the number of x-intercepts of y = 2x 2 – 9x + 5 using the discriminant. a = 2, b = –9, c = 5 Identify the values of a, b, and c. b 2 – 4ac =(–9) 2 – 4(2)(5) = 81 – 40 = 41 Simplify. b 2 – 4ac is positive. Therefore, the function y = 2x 2 – 9x + 5 has two x- intercepts. The graph intercepts the x-axis twice. Substitute 2,–9, and 5 for a, b, and c.

15. 9-9 The Discriminant Additional Example 2B: Using the Discriminant to find the number of x-Intercepts Find the number of x-intercepts of y = 6x 2 – 4x + 5 using the discriminant. a = 6, b = –4, c = 5 Identify the values of a, b, and c. b 2 – 4ac =(–4) 2 – 4(6)(5) = 16 – 120 = –104 Simplify. b 2 – 4ac is negative. Therefore, the function y = 6x 2 – 4x + 5 has no x- intercepts. The graph does not intercept the x-axis. Substitute 6, –4, and 5 for a, b, and c.

16. 9-9 The Discriminant Check It Out! Example 2a Find the number of x-intercepts of y = 5x 2 + 3x + 1 by using the discriminant. a = 5, b = 3, c = 1 Identify the values of a, b, and c. b 2 – 4ac =3 2 – 4(5)(1) = 9 – 20 = –11 Simplify. b 2 – 4ac is negative. Therefore, the function y = 5x 2 + 3x + 1 has no x- intercepts. The graph does not intercept the x-axis. Substitute 5, 3, and 1 for a, b, and c.

17. 9-9 The Discriminant Check It Out! Example 2b Find the number of x-intercepts of y = x 2 – 9x + 4 by using the discriminant. a = 1, b = –9, c = 4 Identify the values of a, b, and c. b 2 – 4ac =(–9) 2 – 4(1)(4) = 81 – 16 = 65 Simplify. b 2 – 4ac is positive. Therefore, the function y = x 2 – 9x + 4 has two x- intercepts. The graph intercepts the x-axis twice. Substitute 1, –9, and 4 for a, b, and c.

18. 9-9 The Discriminant The ringer on a carnival strength test is 2 feet off the ground and is shot upward with an initial velocity of 30 feet per second. Will it reach a height of 20 feet? Use the discriminant to explain your answer. Additional Example 3: Physical Science Application The height h in feet of an object shot straight up with initial velocity v in feet per second is given by h = –16t 2 + vt + c, where c is the initial height of the object above the ground.

19. 9-9 The Discriminant Additional Example 3 Continued h = –16t 2 + vt + c 20 = –16t 2 + 30t + 2 0 = –16t 2 + 30t + (–18) b 2 – 4ac 30 2 – 4(–16)(–18) = –252 Substitute 20 for h, 30 for v, and 2 for c. Subtract 20 from both sides. Evaluate the discriminant. Substitute – 16 for a, 30 for b, and – 18 for c. The discriminant is negative, so there are no real solutions. The ringer will not reach a height of 20 feet.

20. 9-9 The Discriminant If the object is shot straight up from the ground, the initial height of the object above the ground equals 0. Helpful Hint

21. 9-9 The Discriminant Check It Out! Example 4 What if…? Suppose a weight is shot straight up from the ground with an initial velocity of 20 feet per second. Will it reach the height of 45 feet? Use the discriminant to explain your answer. h = –16t 2 + vt + c 45 = –16t 2 + 20t Substitute 45 for h, and 20 for v. 0 = –16t 2 + 20t + (–45) Subtract 45 from both sides. b 2 – 4ac 20 2 – 4(–16)(–45) = –2080 Evaluate the discriminant. Substitute – 16 for a, 20 for b, and – 45 for c. No; for the equation 45 = –16t 2 + 20t, the discriminant is negative, so the weight will not ring the bell.

22. 9-9 The Discriminant 1. Find the number of solutions of 5x 2 – 19x – 8 = 0 by using the discriminant. 2. Find the number of x-intercepts of y = –3x 2 + 2x – 4 by using the discriminant. 3. An object is shot up from 4 ft off the ground with an initial velocity of 48 ft/s. Will it reach a height of 40 ft? Use the discriminant to explain your answer. Lesson Quiz 2 2 The discriminant is 0. The object will reach its maximum height of 40 ft once.