Chapter 3 Quadratic Functions and Equations Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 3.2 Quadratic Equations and Problem Solving ♦ Understand basic concepts about quadratic equations ♦ Use factoring, the square root property, completing the square, and the quadratic formula to solve quadratic equations ♦ Understand the discriminant ♦ Solve problems involving quadratic equations Copyright © 2014, 2010, 2006 Pearson Education, Inc. 2 Quadratic Equation A quadratic equation in one variable is an equation that can be written in the form ax2 + bx + c where a, b, and c are constants with a ≠ 0. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 3 Solving Quadratic Equations Quadratic equations can have no real solutions, one real solution, or two real solutions. The are four basic symbolic strategies in which quadratic equations can be solved. • Factoring • Square root property • Completing the square • Quadratic formula Copyright © 2014, 2010, 2006 Pearson Education, Inc. 4 Factoring Factoring is a common technique used to solve equations. It is based on the zeroproduct property, which states that if ab = 0, then a = 0 or b = 0 or both. It is important to remember that this property works only for 0. For example, if ab = 1, then this equation does not imply that either a = 1 or b = 1. For example, a = 1/2 and b = 2 also satisfies ab = 1 and neither a nor b is 1. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 5 Example: Solving quadratic equations with factoring Solve the quadratic equation 12t2 = t + 1. Check your results. Solution 2 12t t 1 12t t 1 0 2 3t 14t 1 0 3t 1 0 or 4t 1 0 1 t or 3 1 t 4 Copyright © 2014, 2010, 2006 Pearson Education, Inc. 6 Example: Solving quadratic equations with factoring Solution continued Check: 2 1 1 12 1 3 3 2 1 1 12 1 4 4 4 4 3 3 3 3 4 4 Copyright © 2014, 2010, 2006 Pearson Education, Inc. 7 The Square Root Property Let k be a nonnegative number. Then the solutions to the equation x2 = k are given by x k. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 8 Example: Using the square root property If a metal ball is dropped 100 feet from a water tower, its height h in feet above the ground after t seconds is given by h(t) = 100 – 16t2. Determine how long it takes the ball to hit the ground. Solution The ball strikes the ground when the equation 100 – 16t2 = 0 is satisfied. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 9 Example: Using the square root property Solution continued 100 16t 2 0 100 16t 2 100 t 16 2 100 10 t 16 4 The ball strikes the ground after 10/4, or 2.5, seconds. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 10 Completing the Square Another technique that can be used to solve a quadratic equation is completing the square. If a quadratic equation is written in the form x2 + kx =d, where k and d are constants, then the equation can be solved using 2 2 k k x kx x . 2 2 2 Copyright © 2014, 2010, 2006 Pearson Education, Inc. 11 Example: Completing the Square Solve 2x2 – 8x = 7. Solution Divide each side by 2: 7 2 x 4x 2 7 2 x 4x 4 4 2 2 15 x2 2 15 x2 2 15 x 2 2 Copyright © 2014, 2010, 2006 Pearson Education, Inc. 12 Symbolic, Numerical and Graphical Solutions Quadratic equations can be solved symbolically, numerically, and graphically. The following example illustrates each technique for the equation x(x – 2) = 3. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 13 Symbolic Solution Copyright © 2014, 2010, 2006 Pearson Education, Inc. 14 Numerical Solution Copyright © 2014, 2010, 2006 Pearson Education, Inc. 15 Graphical Solution Copyright © 2014, 2010, 2006 Pearson Education, Inc. 16 Quadratic Formula The solutions to the quadratic equation ax2 + bx + c = 0, where a ≠ 0, are given by b b 4ac x . 2a 2 Copyright © 2014, 2010, 2006 Pearson Education, Inc. 17 Example: Using the quadratic formula Solve the equation 3x2 – 6x + 2 = 0. Solution Let a = 3, b = 6, and c = 2. b b 2 4ac x 2a x 6 6 4 3 2 2 3 2 6 12 1 1 x 1 4 3 1 3 6 6 3 Copyright © 2014, 2010, 2006 Pearson Education, Inc. 18 The Discriminant If the quadratic equation ax2 + bx + c = 0 is solved graphically, the parabola y = ax2 + bx + c can intersect the x-axis zero, one, or two times. Each x-intercept is a real solution to the quadratic equation. b 2 4ac 0 b 2 4ac 0 Copyright © 2014, 2010, 2006 Pearson Education, Inc. b 2 4ac 0 19 Quadratic Equations and Discriminant To determine the number of real solutions to ax2 + bx + c = 0 with a ≠ 0, evaluate the discriminant b2 – 4ac. 1. If b2 – 4ac > 0, there are two real solutions. 2. If b2 – 4ac = 0, there is one real solution. 3. If b2 – 4ac < 0, there are no real solutions. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 20 Example: Using the discriminant Use the discriminant to find the number of solutions to 4x2 – 12x + 9 = 0. Then solve the equation by using the quadratic formula. Support your answer graphically. Solution Let a = 4, b = –12, and c = 9 b2 – 4ac = (–12)2 – 4(4)(9) = 0 Discriminant is 0, there is one solution. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 21 Example: Using the discriminant Solution continued b b 4ac x 2a 2 x 12 0 8 3 x 2 The only solution is 3/2. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 22 Example: Using the discriminant Solution continued The graph suggests there is only one intercept 3/2. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 23 Problem Solving Many types of applications involve quadratic equations. To solve these problems, we use the steps for “Solving Application Problems” from Section 2.2. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 24 Example: Solving a construction problem A box is being constructed by cutting 2-inch squares from the corners of a rectangular piece of cardboard that is 6 inches longer than it is wide. If the box is to have a volume of 224 cubic inches, find the dimensions of the piece of cardboard. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 25 Example: Solving a construction problem Solution Step 1: Let x be the width and x + 6 be the length. Step 2: Draw a picture. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 26 Example: Solving a construction problem Since the height times the width times the length must equal the volume, or 224 cubic inches, the following can be written 2(x – 4)(x + 2) = 224 Step 3: Write the quadratic equation in the form ax2 + bx + c = 0 and factor. x 2x 8 112 2 x 2 2x 120 0 x 12x 10 0 x 12 or x 10 Copyright © 2014, 2010, 2006 Pearson Education, Inc. 27 Example: Solving a construction problem The dimensions can not be negative, so the width is 12 inches and the length is 6 inches more, or 18 inches. Step 4: After the 2-inch-square corners are cut out, the dimensions of the bottom of the box are 12 – 4 = 8 inches by 18 – 4 = 14 inches. The volume of the box is then 2•8•14 = 224 cubic inches, which checks. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 28

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