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OCF.01.5 - Finding Zeroes of Quadratic Equations

OCF.01.5 - Finding Zeroes of Quadratic Equations

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OCF.01.5 - Finding Zeroes of Quadratic Equations

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  1. OCF.01.5 - Finding Zeroes of Quadratic Equations MCR3U - Santowski

  2. (A) Review • Zeroes is another term for roots or x-intercepts - basically, the point where the function crosses the x-axis. At this point, the y value of the function is 0. • A quadratic may have one of the following three possibilities : 2 distinct zeroes, one zero (the x-intercept and the vertex are one and the same), or no zeroes (the graph does not cross the x-axis) • See diagrams on the next slide

  3. (A) Review – Zeroes of Quadratics • Diagrams of each scenario (0,1,2 zeroes)

  4. (B) Finding the Zeroes • The zeroes of a quadratic function can be found in a variety of ways: • (i) Factoring: If a quadratic equation can be factored to the form of y = a(x - s)(x - t), then the zeroes are at (s,0) and (t,0) • ex: Find the roots of y = 4x2 - 12x + 9 • (ii) Completing the Square technique: If an equation can be written in the y = a(x – h)2 + k form, then the (x – h)2 term can be isolated in order to solve for x • ex 1. Find the roots of y = x2 - 6x - 27 by using the method of completing the square • ex 2. Solve y > 2x2 - 5x - 1 using the completing the square technique

  5. (B) Graph of y > 2x2 – 5x – 1

  6. (B) Finding the Zeroes • (iii) The Quadratic Formula: • For an equation in the form of y = ax2+ bx+c, then the quadratic formula may be used: • x = [- b +(b2-4ac)] / 2a • ex 1. Find the roots of the y = x2 - 2x - 3 using the quadratic formula • ex 2. Solve y < -2x2 + 5x + 8 using the quadratic formula • (iv) Using a Graphing Calculator/Technology • ex 1. Graph y = 4x2 + 8x - 24 and find the intercepts. • ex 2. Solve y < 1/4x2 + 5x – 9 using the GDC

  7. (C) The Discriminant • You can use part of the quadratic formula, the discriminant (b2 - 4ac) to predict the number of roots a quadratic equation has. • If b2 - 4ac > 0, then the quadratic equation has two zeroes • ex: y = 2x2 + 3x – 6 • If b2 - 4ac = 0, then the quadratic equation has one zero • ex: y = 4x2 + 16x + 16 • If b2 - 4ac < 0, then the quadratic equation has no zeroes • ex: y = -3x2 + 5x - 3

  8. (D) Visualizing and Interpreting The Discriminant

  9. (E) Interpretation of Zeroes • ex 1. The function h = -5t2 + 20t + 2 gives the approximate height, in meters, of a thrown football as a function of time in seconds. The ball hit the ground before the receiver could get there. • (a) For how long was the ball in the air? • (b) For how many seconds was the height of the ball at least 10 meters?

  10. (F) Internet Links • College Algebra Tutorial on Quadratic Equations • Solving Quadratic Equations Lesson - I from Purple Math

  11. (D) Homework • Handout from MHR, p129, Q5bce, 6g, 7, 8ab, 9acegk, 11ace, 12egmn • Nelson Textbook, p325, Q1-5eol, 7-11,6