CHAPTER 9 Quadratic Equations Slide 2Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 9.1Introduction to Quadratic Equations 9.2Solving Quadratic Equations by Completing the Square 9.3The Quadratic Formula 9.4Formulas 9.5Applications and Problem Solving 9.6Graphs of Quadratic Equations 9.7Functions

OBJECTIVES 9.1 Introduction to Quadratic Equations Slide 3Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. aWrite a quadratic equation in standard form ax 2 + bx + c = 0, a > 0, and determine the coefficients a, b, and c. bSolve quadratic equations of the type ax 2 + bx = 0, where b ≠ 0, by factoring. cSolve quadratic equations of the type ax 2 + bx + c = 0, where b ≠ 0 and c ≠ 0, by factoring. dSolve applied problems involving quadratic equations.

The following are quadratic equations. They contain polynomials of second degree. 4x 2 + 7x – 5 = 0 3y 2 – y = 9 5a 2 = 8a12m 2 = Introduction to Quadratic Equations a Write a quadratic equation in standard form ax 2 + bx + c = 0, a > 0, and determine the coefficients a, b, and c. Slide 4Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

A quadratic equation is an equation equivalent to an equation of the type ax 2 + bx + c = 0, a > 0, where a, b, and c are real-number constants. We say that the preceding is the standard form on a quadratic equation. 9.1 Introduction to Quadratic Equations Quadratic Equation Slide 5Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE a. 5x 2 + 8x – 3 = 0 The equation is in standard form. 5x 2 + 8x – 3 = 0 a = 5; b = 8; c = –3 9.1 Introduction to Quadratic Equations a Write a quadratic equation in standard form ax 2 + bx + c = 0, a > 0, and determine the coefficients a, b, and c. AWrite in standard form and determine a, b, and c. Slide 6Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE b.6y 2 = 5y 6y 2 – 5y = 0 a = 6; b = –5; c = Introduction to Quadratic Equations a Write a quadratic equation in standard form ax 2 + bx + c = 0, a > 0, and determine the coefficients a, b, and c. AWrite in standard form and determine a, b, and c. Slide 7Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE Solution 8x 2 + 3x = 0 x(8x + 3) = 0Factoring x = 0 or 8x + 3 = 0 Using the principle of zero products x = 0 or 8x = –3 x = 0 or 9.1 Introduction to Quadratic Equations b Solve quadratic equations of the type ax 2 + bx = 0, where b ≠ 0, by factoring. BSolve: 8x 2 + 3x = 0. Slide 8Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE Check: 8x 2 + 3x = 0 8(0) 2 + 3(0) = 0 0 = 0 True Both solutions check. 9.1 Introduction to Quadratic Equations b Solve quadratic equations of the type ax 2 + bx = 0, where b ≠ 0, by factoring. BSolve: 8x 2 + 3x = 0. Slide 9Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 0 = 0 True 8x 2 + 3x = 0

A quadratic equation of the type ax 2 + bx = 0, where c = 0 and b ≠ 0, will always have 0 as one solution and a nonzero number as the other solution. 9.1 Introduction to Quadratic Equations b Solve quadratic equations of the type ax 2 + bx = 0, where b ≠ 0, by factoring. Slide 10Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE Solution : Write the equation in standard form and then try factoring. (y – 7)(y – 2) = 4y – 22 y 2 – 9y + 14 = 4y – 22 Multiplying y2 – 13y + 36 = 0 Standard form (y – 4)(y – 9) = 0 y – 4 = 0 or y – 9 = 0 y = 4 or y = 9 The solutions are 4 and Introduction to Quadratic Equations c Solve quadratic equations of the type ax 2 + bx + c = 0, where b ≠ 0 and c ≠ 0, by factoring. CSolve: (y – 7)(y – 2) = 4y – 22 Slide 11Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE The number of diagonals d in a polygon that has n sides is given by the formula If a polygon has 54 diagonals, how many sides does it have? 9.1 Introduction to Quadratic Equations d Solve applied problems involving quadratic equations. DApplications of Quadratic Equations Slide 12Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE 1. Familiarize. A sketch can help us to become familiar with the problem. We draw a hexagon (6 sides) and count the diagonals. As the formula predicts, for n = 6, there are 9 diagonals: 9.1 Introduction to Quadratic Equations d Solve applied problems involving quadratic equations. DApplications of Quadratic Equations Slide 13Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE 2. Translate. Since the number of diagonals is 54, we substitute 54 for d: 9.1 Introduction to Quadratic Equations d Solve applied problems involving quadratic equations. DApplications of Quadratic Equations (continued) Slide 14Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE 3. Solve. We solve the equation for n, first reversing the equation for convenience. 9.1 Introduction to Quadratic Equations d Solve applied problems involving quadratic equations. DApplications of Quadratic Equations (continued) Slide 15Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE 4. Check. Since the number of sides cannot be negative, –9 cannot be a solution. 5. State. The polygon has 12 sides. 9.1 Introduction to Quadratic Equations d Solve applied problems involving quadratic equations. DApplications of Quadratic Equations Slide 16Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.