1 OCF 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 = 4x x + 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 = x 2 - 6x - 27 by using the method of completing the square  ex 2. Solve y > 2x 2 - 5x - 1 using the completing the square technique  ex 2. Solve y > 2x 2 - 5x - 1 using the completing the square technique

5 (B) Graph of y > 2x 2 – 5x – 1

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

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

8 (D) Visualizing and Interpreting The Discriminant

9 (E) Interpretation of Zeroes  ex 1. The function h = -5t t + 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 College Algebra Tutorial on Quadratic Equations College Algebra Tutorial on Quadratic Equations  Solving Quadratic Equations Lesson - I from Purple Math Solving Quadratic Equations Lesson - I from Purple Math 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