IB Math Analysis & Approaches SL Linear and Quadratic Functions PDF

Chapter 3: Linear and Quadratic Functions 3.6 The quadratic formula and the discriminant Essential Questions: How can the discriminant be used to determine the nature and number of solutions to a quadratic equation? The Quadratic Formula ax 2 + bx + c = 0...

Chapter 3: Linear and Quadratic Functions 3.6 The quadratic formula and the discriminant Essential Questions: How can the discriminant be used to determine the nature and number of solutions to a quadratic equation? The Quadratic Formula ax 2 + bx + c = 0 Solve by completing the square. Divide both sides of the equation by a. c Subtract a from both sides of the equation. b 2 b 2 x 2 + ba x + = − ca +

2a 2a b2 b 2 −4ac x 2 + ba x + 4a2 = 4a2 Factorize the left-hand side of the equation. √ b b 2 −4ac x+ 2a =± 2a Solve for x. IB Math Analysis & Approaches SL 1/13 The Quadratic Formula The Quadratic Formula The solutions to any quadratic equation given in the form ax 2 + bx + c = 0, where a 6= 0, are given by the quadratic formula: √ −b ± b 2 − 4ac x= 2a −b 1. What is the significance of the term 2a ? √ b 2 −4ac 2. What is the significance of the term 2a ? IB Math Analysis & Approaches SL 2/13 Solving Quadratic Equations Use the quadratic formula to solve the following equations. Describe the nature of the roots. 1. 3x 2 + 6x = −2 2. 6x 2 = 7x + 5 3. x 2 − 4x + 5 = 0 IB Math Analysis & Approaches SL 3/13 Textbook Exercises Textbook Exercises To practice and apply these concepts, work through the following exercises: Exercise 3U p.166 #1-4 IB Math Analysis & Approaches SL 4/13 The Discriminant The Discriminant A quadratic equation ax 2 + bx + c = 0 has discriminant ∆ = b 2 − 4ac. The symbol ∆ is used to represent the discriminant. For... The equation ax 2 + bx + c = 0 The graph of y = ax 2 + bx + c has has how many roots? how many x-intercepts? b 2 −4ac > 0 b 2 −4ac = 0 b 2 −4ac < 0 IB Math Analysis & Approaches SL 5/13 The Discriminant Use the discriminant to describe the nature of the roots of the quadratic equation. 1. x 2 + x + 5 = 0 2. 16x 2 − 8x + 1 = 0 3. 6x 2 − 47x − 8 = 0 IB Math Analysis & Approaches SL 6/13 The Discriminant Find the values of k for which the graph of y = x 2 + 9x + k has two distinct real roots. IB Math Analysis & Approaches SL 7/13 Textbook Exercises Textbook Exercises To practice and apply these concepts, work through the following exercises: Exercise 3V p.169 #1-4 IB Math Analysis & Approaches SL 8/13 Solving Quadratic Equations Use any method to solve the following quadratic equations. Which method is most efficient? 1. x 2 + 10x + 25 = 0 2. x 2 − 4x + 11 = 0 3. x 2 + 5x + 6 = 0 4. 5x 2 − 6x + 9 = 0 5. 2x 2 − 3x + 17 = 0 IB Math Analysis & Approaches SL 9/13 Solving Quadratic Inequalities Solve 2x 2 − 5 ≥ x − 3. IB Math Analysis & Approaches SL 10/13 Solving Quadratic Equations Find the values of k for which the graph of y = x 2 + kx + 9 has no x-intercepts. IB Math Analysis & Approaches SL 11/13 Textbook Exercises Textbook Exercises To practice and apply these concepts, work through the following exercises: Exercise 3W p.172 #1-6 IB Math Analysis & Approaches SL 12/13  Assignment: 1. Process your notes Chunk the information by numbering each big idea Circle key words and new vocabulary Highlight main ideas Fill in any gaps and add clarifying comments Write questions next to any points of confusion 2. Textbook Exercises: Read pp.164-172 3U p.166 #2d, 2e, 2f, 2g 3V p.169 #2, 3, 4 3W p.172 #1, 6 IB Math Analysis & Approaches SL 13/13