Copyright © 2007 Pearson Education, Inc. Slide 3-1.

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Copyright © 2007 Pearson Education, Inc. Slide 3-1

Copyright © 2007 Pearson Education, Inc. Slide 3-2 Chapter 3: Polynomial Functions 3.1 Complex Numbers 3.2 Quadratic Functions and Graphs 3.3 Quadratic Equations and Inequalities 3.4 Further Applications of Quadratic Functions and Models 3.5 Higher Degree Polynomial Functions and Graphs 3.6 Topics in the Theory of Polynomial Functions (I) 3.7 Topics in the Theory of Polynomial Functions (II) 3.8 Polynomial Equations and Inequalities; Further Applications and Models

Copyright © 2007 Pearson Education, Inc. Slide 3-3 3.3 Quadratic Equations and Inequalities Consider the –zeros of –x-intercepts of –solution set of Each is solved by finding the numbers that make Quadratic Equation in One Variable An equation that can be written in the form ax 2 + bx + c = 0 where a, b, and c are real numbers with a  0, is a quadratic equation in standard form.

Copyright © 2007 Pearson Education, Inc. Slide 3-4 3.3 Zero-Product Property If a and b are complex numbers and ab = 0, then a = 0 or b = 0 or both equal zero. Example Solve Solution The solution set of the equation is {– 4,2}. Note that the zeros of P(x) = 2x 2 + 4x – 16 are –4 and 2, and the x-intercepts are (– 4,0) and (2,0).

Copyright © 2007 Pearson Education, Inc. Slide 3-5 3.3 Example of a Quadratic with One Distinct Zero Example Solve Solution Graphing Calculator Solution There is one distinct zero, 3. It is sometimes called a double zero, or double solution (root) of the equation.

Copyright © 2007 Pearson Education, Inc. Slide 3-6 3.3 Solving x 2 = k Square Root Property The solution of is Figure 26 pg 3-38

Copyright © 2007 Pearson Education, Inc. Slide 3-7 3.3 Examples Using the Square Root Property Solve each equation. (a) (b) Graphing Calculator Solution (a) (b) This solution is an approximation.

Copyright © 2007 Pearson Education, Inc. Slide 3-8 3.3 The Quadratic Formula Complete the square on and rewrite P in the form To find the zeros of P, use the square root property to solve for x.

Copyright © 2007 Pearson Education, Inc. Slide 3-9 The expression under the radical, is called the discriminant. The discriminant determines whether the quadratic equation has –two real solutions if –one real solution if –no real solutions if 3.3 The Quadratic Formula and the Discriminant The Quadratic Formula The solutions of the equation, where, are

Copyright © 2007 Pearson Education, Inc. Slide 3-10 3.3 Solving Equations with the Quadratic Formula Solve Analytic Solution Rewrite in the form ax 2 + bx + c = 0 Substitute into the quadratic formula a = 1, b = – 4, and c = 2, and simplify.

Copyright © 2007 Pearson Education, Inc. Slide 3-11 3.3 Solving Equations with the Quadratic Formula Graphical Solution With the calculator, we get an approximation for

Copyright © 2007 Pearson Education, Inc. Slide 3-12 3.3 Solving a Quadratic Equation with No Solution Example Solve Analytic Solution Graphing Calculator Solution Figure 30 pg 3-43

Copyright © 2007 Pearson Education, Inc. Slide 3-13 Similar for a quadratic function P(x) with a < 0. 3.3 Possible Orientations for Quadratic Function Graphs (a > 0)

Copyright © 2007 Pearson Education, Inc. Slide 3-14 3.3 Solving a Quadratic Inequality Analytically Solve The function is 0 when x = – 3 or x = 4. It will be greater than 0 or less than zero for any other value of x. So we use a sign graph to determine the sign of the product (x + 3)(x – 4). For the interval (–3,4), one factor is positive and the other is negative, giving a negative product. The product is positive elsewhere since (+)(+) and (–)(–) is positive.

Copyright © 2007 Pearson Education, Inc. Slide 3-15 3.3 Solving a Quadratic Inequality Graphically Verify the analytic solution to Y 1 < 0 in the interval (–3,4). Solving a Quadratic Inequality 1.Solve the corresponding quadratic equation. 2.Identify the intervals determined by its solutions. 3.Use a sign graph to determine the solution interval(s). 4.Decide whether or not the endpoints are included.

Copyright © 2007 Pearson Education, Inc. Slide 3-16 3.3 Literal Equations Involving Quadratics Example Solve (a) (b) Solution (a) (b)

Copyright © 2007 Pearson Education, Inc. Slide 3-17 3.3 Modeling Length of Life The survival rate after age 65 is approximated by where x is measured in decades. This function gives the probability that an individual who reaches age 65 will live at least x decades (10x years) longer. Find the age for which the survival rate is.5 for people who reach the age of 65. Interpret the result. SolutionLet Y 1 = S(x), Y 2 =.5, and find the positive x-value of an intersection point. Discard x = – 3. For x  2.2, half the people who reach age 65 will live about 2.2 decades or 22 years longer to age 87.