Math 20-1 Chapter 4 Quadratic Equations General Outcome: Develop algebraic and graphical reasoning through the study of relations. Specific Outcomes: RF1. Factor polynomial expressions of the form: ax 2 + bx + c, a 0 a 2 x 2 b 2 y 2, a 0, b 0 a(f(x)) 2 + b(f(x)) + c, a 0 a 2 (f(x)) 2 b 2 (g(y)) 2, a 0, b 0 where a, b and c are rational numbers. [CN, ME, R] RF4. Analyze quadratic functions of the form y = ax 2 + bx + c to identify characteristics of the corresponding graph, including: vertex domain and range direction of opening axis of symmetry x- and y-intercepts and to solve problems. [CN, PS, R, T, V] [ICT: C6 4.1, C6 4.3] RF5. Solve problems that involve quadratic equations. [C, CN, PS, R, T, V] [ICT: C6 4.1] Mark Assignments 4.1: Page 215 217 # 1 5, 9, 11, 17, 18 4.2: Page 229 233 # 1-2(ac), 3-4(bd), 5-7, 9-10(abdf), 30 4.3: Page 240 243 # 2-7(acf), 18. 4.4: Page 254 257 # (3, 4, 5, 7, 21-23) (1 & 2) Quiz 4 Chapter 4 Test Date: Date:
4.1 Graphical Solutions of Quadratic Equations Getting Started: Solve this linear equation: 4x 10 = 4 3x QUADRATIC FUNCTION A Quadratic Function is a function that can be written in the form: f(x) = ax 2 + bx + c, where a, b, c R and a 0 or in equation form as: y = ax 2 + bx + c, where a, b, c R and a 0 QUADRATIC EQUATION A Quadratic Equation is an equation that can be written in the form: ax 2 + bx + c = 0, where a, b, c R and a 0 SIMPLY PUT IT IS A QUADRATIC FUNCTION THAT IS EQUAL TO A SPECIFIC VALUE!! In this chapter, we are going to learn methods of solving quadratic equations!
YOU TRY Given the equation 2x 2 + 12x + 16 = 0, what could you do to find the value(s) for x that solve this equation?? (HINT: Where does 2x 2 + 12x + 16 equal zero??) NOTE: The solutions to a quadratic equation are called the roots of the equation! The graph of f(x) = x 2 + 4x + 3 is shown. Fill in the blanks! The GRAPH of f(x) = x 2 + 4x + 3 has x-intercepts x = and x =. The FUNCTION f(x) = x 2 + 4x + 3 has zeros and. The EQUATION x 2 + 4x + 3 = 0 has roots (or solutions) x = and x =
How many x-intercepts can the graph of a quadratic function have? Since these x-intercepts (zeros) represent the solutions to the corresponding quadratic equation, how many solutions (roots) can a quadratic equation have? Example 1 The flight path of a water skier jumper off of a ramp is modelled by the equation: h(t) = 5.0 + 24.46t - 4.9t 2 where h(t) is the height of the jumper in meters and t is the time in seconds. How long was he in the air for? Solve using your calculator. When is the water-skier 15 m in the air?
Example 2 EXAMPLE 3: SOLVING QUADRATIC EQUATIONS GRAPHICALLY Determine the roots of each quadratic equation? If necessary, round answers to the nearest tenth. a) x 2 + 6x + 3 = 0 b) x 2 = 2x + 1
c) 2x 2 + 22 = 12x EXAMPLE 4: CHECKING THE SOLUTIONS TO A QUADRATIC EQUATION Solve each equation. Check your solutions. Note: Check the solution(s) by substituting them into the ORIGINAL equation. a) 2x 2 + 5x 3 = 0 b) x 2 6x = 9
EXAMPLE 5: SOLVE A PROBLEM INVOLVING QUADRATIC EQUATIONS When a basketball is thrown, its height can be modeled by the function h(t) = 4.9t 2 + 15t + 1, where h is the height of the ball, in metres, and t is time, in seconds. a) What does the equation 0 = 4.9t 2 + 15t + 1 represent in this situation? b) How long was the basketball in the air?
Review: 4.2 Factoring Quadratic Equations Factor each of the following quadratic expressions: 1. x 2 9x + 18 6. 1 4 x2 + 1 2 x 6 2. t 2 + t 56 7. 0.5z 2 5.4z + 4 3. 5a 2 5a 30 8. 9x 2 0.25y 2 4. 3n 2 11n 4 9. 1 4 d2 1 49 f2 5. 2x 2 17x + 30 10. 8a 2 18b 2
Extending What You Know: Factor each of the following quadratic expressions: 1. 9(x 1) 2 100y 2 3. 4(s 2) 2 + 9(2t + 1) 2 2. 4(x + 3) 2 + 8(x + 3) 5 4. 2(x + 2) 2 + 12(x + 2) + 9 KEY IDEAS: Last class we learned one strategy for solving a quadratic is graphing. Another strategy is to use FACTORING!! The zero product property states that if the product of two real numbers is zero, then one or both of the numbers must equal zero. ie: This means if xy = 0, then This principle applies to quadratic equations. ex: x 2 + 2x 24 = 0 (x + 6)(x 4) = 0
EXAMPLE 1: SOLVING FACTORED EQUATIONS Solve each equation. Note: They are already factored for you!! a) (x 5)(x + 9) = 0 b) (2x + 9)(x 4) = 0 c) (x + 3 4 ) (x 11 2 ) = 0 d) x(3x 14) = 0
EXAMPLE 2: SOLVING QUADRATIC EQUATIONS BY FACTORING Determine the roots of each quadratic equation. Verify/check your answers. Note: Check the solution(s) by substituting them into the ORIGINAL equation. d) x 2 + 4x 21 = 0 e) x 2 = 10x 16 f) 2x 2 = 9x + 5
EXAMPLE 3: SOLVING MORE QUADRATIC EQUATIONS BY FACTORING Solve each equation. c) 2x 2 + 18 = 12x d) 8x 2 18 = 0 e) 2x 2 = 4x f) (2x 3)(x + 1) = 3
g) 2x(x 6) + 3x = 2x 9 4.3 Solving Quadratic Equations by Completing the Square KEY IDEAS: We can solve quadratic equations in vertex form y = a(x p) 2 + q by isolating the squared term and taking the square root of both sides. THE SQUARE ROOT OF A NUMBER CAN BE POSITIVE OR NEGATIVE SO THERE ARE TWO POSSIBLE SOLUTIONS TO THESE EQUATIONS!! EXAMPLE 1: TAKING SQUARE ROOTS TO SOLVE QUADRATIC EQUATIONS Solve each equation. Verify your solutions. e) x 2 = 36 f) (x 4) 2 64 = 0
EXAMPLE 2: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE Solve each quadratic equation by completing the square. Express your answers as exact roots. g) a 2 + 18a + 32 = 0 h) t 2 8t 4 = 0 i) 4s 2 + 48s + 141 = 0
EXAMPLE 3: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE Determine the roots of 2x 2 3x + 7 = 0 to the nearest hundredth. Then, use technology to verify your answers. Why is it sometime more useful to complete the square then it is to solve an equation graphically or by factoring?
4.4 The Quadratic Formula Developing the Quadratic Formula: 1. Complete the square to solve the following quadratic equation: 2x 2 + 7x + 1 = 0 2. Complete the square using the general quadratic equation in standard form: ax 2 + bx + c = 0
KEY IDEAS: The Quadratic Formula: x = b ± b2 4ac 2a The quadratic formula can be used to solve any quadratic equation in general form: ax 2 + bx + c = 0, a 0. EXAMPLE 1: USING THE QUADRATIC FORMULA TO SOLVE EQUATIONS Use the quadratic equation to solve each equation. Express your answers as exact answers and to the nearest hundredth. g) 9x 2 + 12x + 4 = 0
h) x 2 + 6x = 1 i) 3y 2 = 8y 3
EXAMPLE 2: SELECT A STRATEGY TO SOLVE A QUADRATIC EQUATION Solve 6x 2 14x + 8 = 0 using each of the following methods: GRAPHING: FACTORING: COMPLETING THE SQUARE: THE QUADRATIC FORMULA: Which of the above methods gives exact answers? When deciding how to solve a quadratic: 1. Try factoring first!! (It s the quickest method and gives an exact answer!) 2. If factoring is not possible, complete the square or use the quadratic formula. (Either one will give you an exact answer YOUR CHOICE!) 3. Use graphing as a way of checking all your answers!
4.4 Part B The Discriminant KEY IDEAS: The Quadratic Formula: x = b ± b2 4ac 2a The quadratic formula can be used to solve any quadratic equation in general form: ax 2 + bx + c = 0, a 0. The Discriminant is the expression under the radical sign in the quadratic formula: b 2 4ac The discriminant can be useful is providing us some key information about the number of solutions or the nature of the roots of a quadratic equation. How?
Try solving each quadratic equation using the quadratic formula 1. x 2 5x + 4 = 0 What does b 2 4ac equal? How many solutions/roots does this equation have? 2. 8x 2 + x + 17 = 0 What does b 2 4ac equal? How many solutions/roots does this equation have? 3. x 2 + 4x + 4 = 0 What does b 2 4ac equal? How many solutions/roots does this equation have?
Value of the Discriminant b 2 4ac > 0 Nature of the Roots Graph of the Function b 2 4ac < 0 b 2 4ac = 0 EXAMPLE: USING THE DISCRIMINANT TO DETERMINE THE NATURE OF THE ROOTS Use the discriminate to determine the nature of the roots. Check by graphing. a) 2x 2 + 3x + 8 = 0 b) 3x 2 + 9 5x = 0 c) 1 4 n2 + 9 = 3n
EXAMPLE 1: 4.2 4.4 Quadratic Formula Problem Solving Dock jumping is an exciting event in which dogs compete for the longest jumping distance from a dock into a body of water. The path of a terrier on a particular jump can be approximated by the quadratic function h(d) = 3 10 d2 + 11 d + 2, where h is the height above the water and d is the horizontal distance the dog travels from the base of the dock. Both measurements are in feet. Determine the horizontal distance of the jump. 10 EXAMPLE 2: The length of a field is 10 m less than twice the width. The area is 6 600 m 2. Determine the dimensions of the field.
EXAMPLE 3: A 42 widescreen television has a width that is 16 inches more than the height of the screen. Determine the dimensions of the screen to the nearest tenth of an inch. EXAMPLE 4: Leah wants to frame a painting measuring 50 cm by 60 cm. Before framing, she places a mat so that a uniform strip of the mat shows on all sides of the painting. The area of the mat is twice the area of the painting. a) How wide is the strip of exposed mat (showing on all sides of the painting), to the nearest tenth of a centimetre? b) What are the dimensions of the painting (and mat) to be framed?