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Quadratic Formula and Discriminant for California Standards

Learn how to use the quadratic formula and evaluate the discriminant to solve quadratic equations and determine the number of solutions. California Standards-aligned lesson.

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Quadratic Formula and Discriminant for California Standards

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  1. Preview Warm Up California Standards Lesson Presentation

  2. Warm Up • Use the Quadratic Formula to solve each equation. • 1. x2 – 5x – 6 = 0 • 2. 2x2 + 2x – 24 = 0 • 3. x2 + 10x + 25 = 0 1, –6 3, –4 0, –5

  3. 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. Vocabulary discriminant

  5. 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. If the quadratic equation is in standard form, its discriminant is b2 – 4ac. Notice that this is the expression under the square root in the Quadratic Formula.

  6. b2 – 4ac = (–2)2 – 4(3)(2) Additional Example 1A: Using the Discriminant Find the number of solutions of 3x2 – 2x + 2 = 0 by using the discriminant. a = 3, b = –2, c = 2 Identify the values of a, b, and c. Substitute 3, –2, and 2 for a, b, and c. Simplify. = 4 – 24 = –20 b2 – 4ac is negative. There are no real solutions.

  7. b2 – 4ac = 112 – 4(2)(12) Additional Example 1B: Using the Discriminant Find the number of solutions of 2x2 + 11x + 12 = 0 by using the discriminant. a = 2, b = 11, c = 12 Identify the values of a, b, and c. Substitute 2, 11, and 12 for a, b, and c. = 121 – 96 Simplify. = 25 b2 – 4ac is positive. There are two solutions.

  8. b2 – 4ac = 82 – 4(1)(16) Additional Example 1C: Using the Discriminant Find the number of solutions of x2 + 8x + 16 = 0 by using the discriminant. Identify the values of a, b, and c. a = 1, b = 8, c = 16 Substitute 1, 8, and 16 for a, b, and c. = 64 – 64 Simplify. = 0 b2 – 4ac is zero. There is one solution.

  9. b2 – 4ac = (–2)2 – 4(2)(3) Check It Out! Example 1a Find the number of solutions of 2x2 – 2x + 3 = 0 using the discriminant. Identify the values of a, b, and c. a = 2, b = –2, c = 3 Substitute 2, –2, and 3 for a, b, and c. = 4 – 24 Simplify. = –20 b2 – 4ac is negative. There are no real solutions.

  10. b2 – 4ac = 42 – 4(1)(4) Check It Out! Example 1b Find the number of solutions of x2 + 4x + 4 = 0 using the discriminant. Identify the values of a, b, and c. a = 1, b = 4, c = 4 Substitute 1, 4, and 4 for a, b, and c. = 16 – 16 Simplify. = 0 b2 – 4ac is zero. There is one real solution.

  11. 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.

  12. b2 – 4ac = (–9)2 – 4(2)(5) Additional Example 2A: Using the Discriminant to Find the number of x-Intercepts Find the number of x-intercepts of y = 2x2 – 9x + 5 using the discriminant. a = 2, b = –9, c = 5 Identify the values of a, b, and c. Substitute 2,–9, and 5 for a, b, and c. = 81 – 40 Simplify. = 41 b2 – 4ac is positive. Therefore, the function y = 2x2 – 9x + 5 has two x-intercepts. The graph intercepts the x-axis twice.

  13. b2 – 4ac = (–4)2 – 4(6)(5) Additional Example 2B: Using the Discriminant to find the number of x-Intercepts Find the number of x-intercepts of y = 6x2 – 4x + 5 using the discriminant. a = 6, b = –4, c = 5 Identify the values of a, b, and c. Substitute 6, –4, and 5 for a, b, and c. = 16 – 120 Simplify. = –104 b2 – 4ac is negative. Therefore, the function y = 6x2 – 4x + 5 has no x-intercepts. The graph does not intercept the x-axis.

  14. b2 – 4ac = 32 – 4(5)(1) Check It Out! Example 2a Find the number of x-intercepts of y = 5x2 + 3x + 1 by using the discriminant. a = 5, b = 3, c = 1 Identify the values of a, b, and c. Substitute 5, 3, and 1 for a, b, and c. = 9 – 20 Simplify. = –11 b2 – 4ac is negative. Therefore, the function y = 5x2 + 3x + 1 has no x-intercepts. The graph does not intercept the x-axis.

  15. b2 – 4ac = (–9)2 – 4(1)(4) Check It Out! Example 2b Find the number of x-intercepts of y = x2 – 9x + 4 by using the discriminant. a = 1, b = –9, c = 4 Identify the values of a, b, and c. Substitute 1, –9, and 4 for a, b, and c. = 81 – 16 Simplify. = 65 b2 – 4ac is positive. Therefore, the function y = x2 – 9x + 4 has two x-intercepts. The graph intercepts the x-axis twice.

  16. Additional Example 3: Physical ScienceApplication The height h in feet of an object shot straight up with initial velocity v in feet per second is given by h = –16t2 + vt + c, where c is the initial height of the object above the ground. 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.

  17. Additional Example 3 Continued h = –16t2 + vt + c Substitute 20 for h, 30 for v, and 2 for c. 20 = –16t2 + 30t + 2 0 = –16t2 + 30t + (–18) Subtract 20 from both sides. Evaluate the discriminant. b2 – 4ac Substitute –16 for a, 30 for b, and –18 for c. 302 – 4(–16)(–18) = –252 The discriminant is negative, so there are no real solutions. The ringer will not reach a height of 20 feet.

  18. Helpful Hint If the object is shot straight up from the ground, the initial height of the object above the ground equals 0.

  19. No; for the equation 45 = –16t2 + 20t, the discriminant is negative, so the weight will not ring the bell. 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 = –16t2 + vt + c 45 = –16t2 + 20t Substitute 45 for h, and 20 for v. Subtract 45 from both sides. 0 = –16t2 + 20t + (–45) b2 – 4ac Evaluate the discriminant. Substitute –16 for a, 20 for b, and –45 for c. 202 – 4(–16)(–45) = –2080

  20. Lesson Quiz 1. Find the number of solutions of 5x2 – 19x – 8 = 0 by using the discriminant. 2. Find the number of x-intercepts of y = –3x2 + 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. 2 2 The discriminant is 0. The object will reach its maximum height of 40 ft once.

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