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2 8 CHAPTER 8 Quadratic Equations and Functions EXAMPLE Solve for. Solution This equation is in standard form, so a 3, b, and c 5. Substitute these values into the quadratic formula. -b { b - ac a - { - 35 # 3 - { { { or Quadratic formula Use a 3, b, and c The solutions are - 3 and -5, or the solution set is e -, -5 f Solve for. Helpful Hint To replace a, b, and c correctly in the quadratic formula, write the quadratic equation in standard form a + b + c 0. EXAMPLE Solve: - 3. Solution First write the equation in standard form by subtracting 3 from both sides Now a, b -, and c - 3. Substitute these values into the quadratic formula. The solutions are + 0 -b { b - ac a - - { # { + { 0 { 0 { 0 # { 0 and - 0, or the solution set is e - 0, + 0 f. Solve: 3-8. Helpful Hint To simplify the epression { 0 in the preceding eample, note that is factored out of both terms of the numerator before simplifying. { 0 { 0 # { 0

4 88 CHAPTER 8 Quadratic Equations and Functions where f 0 or where Since this equation has real roots, the graph has -intercepts. Similarly, since the equation has no real roots, the graph of f has no -intercepts. -intercepts y y ( 5, 0) ( a, 0) 5 f() f() no -intercept Using the Discriminant In the quadratic formula, -b { b - ac, the radicand b - ac is called the a discriminant because, by knowing its value, we can discriminate among the possible number and type of solutions of a quadratic equation. Possible values of the discriminant and their meanings are summarized net. Discriminant The following table corresponds the discriminant b - ac of a quadratic equation of the form a + b + c 0 with the number and type of solutions of the equation. b ac Positive Zero Negative Number and Type of Solutions Two real solutions One real solution Two comple but not real solutions EXAMPLE 5 Use the discriminant to determine the number and type of solutions of each quadratic equation. a b c Solution a. In + + 0, a, b, and c. Thus, b - ac - 0 Since b - ac 0, this quadratic equation has one real solution. b. In this equation, a 3, b 0, c. Then b - ac Since b - ac is negative, the quadratic equation has two comple but not real solutions. c. In this equation, a, b -7, and c -. Then b - ac Since b - ac is positive, the quadratic equation has two real solutions. 5 Use the discriminant to determine the number and type of solutions of each quadratic equation. a b c

5 Section 8. Solving Quadratic Equations by the Quadratic Formula 89 The discriminant helps us determine the number and type of solutions of a quadratic equation, a + b + c 0. Recall from Section 5.8 that the solutions of this equation are the same as the -intercepts of its related graph f( a + b + c. This means that the discriminant of a + b + c 0 also tells us the number of -intercepts for the graph of f a + b + c or, equivalently, y a + b + c. Graph of f a b c or y a b c b ac 0, f () has two -intercepts y b ac 0, f () has one -intercept y b ac 0, f () has no -intercepts y 3 Solving Problems Modeled by Quadratic Equations The quadratic formula is useful in solving problems that are modeled by quadratic equations. EXAMPLE Calculating Distance Saved At a local university, students often leave the sidewalk and cut across the lawn to save walking distance. Given the diagram below of a favorite place to cut across the lawn, approimate how many feet of walking distance a student saves by cutting across the lawn instead of walking on the sidewalk. 50 ft 0 Solution. UNDERSTAND. Read and reread the problem. In the diagram, notice that a triangle is formed. Since the corner of the block forms a right angle, we use the Pythagorean theorem for right triangles. You may want to review this theorem.. TRANSLATE. By the Pythagorean theorem, we have In words: leg + leg hypotenuse Translate: SOLVE. Use the quadratic formula to solve Square + 0 and Set the equation equal to Divide by.

6 90 CHAPTER 8 Quadratic Equations and Functions Here, a, b 0, c By the quadratic formula, -0 { # -0 { { 00 # - 0 { 5 Simplify. -0 { 00-0 { 0. INTERPRET Check: Your calculations in the quadratic formula. The length of a side of a triangle can t be negative, so we reject Since feet, the walking distance along the sidewalk is feet. 5 ft 3 State: A student saves about 8-50 or 8 feet of walking distance by cutting across the lawn. Given the diagram, approimate to the nearest foot how many feet of walking distance a person can save by cutting across the lawn instead of walking on the sidewalk. EXAMPLE 7 Calculating Landing Time An object is thrown upward from the top of a 00-foot cliff with a velocity of feet per second. The height h in feet of the object after t seconds is h -t + t + 00 How long after the object is thrown will it strike the ground? Round to the nearest tenth of a second. 00 ft Solution. UNDERSTAND. Read and reread the problem.. TRANSLATE. Since we want to know when the object strikes the ground, we want to know when the height h 0, or 0 -t + t SOLVE. First we divide both sides of the equation by -. 0 t - 3t - 50 Divide both sides by -. Here, a, b -3, and c -50. By the quadratic formula, t - -3 { # 3 { { 809 8

8 9 CHAPTER 8 Quadratic Equations and Functions 8. Eercise Set Use the quadratic formula to solve each equation. These equations have real number solutions only. See Eamples through 3.. m + 5m - 0. p + p y 5y y + 0y y + 5y m - m 7 0. n - 9n. 3m - 7m y + 5 y y y + 8. y y y - 8 y. m + m - 5m pp - + p + 3 MIXED Use the quadratic formula to solve each equation. These equations have real solutions and comple but not real solutions. See Eamples through y + 0y y + y y + 5 y y y n - n 0. ap - b p Use the discriminant to determine the number and types of solutions of each equation. See Eample Solve. See Eamples 7 and Nancy, Thelma, and John Varner live on a corner lot. Often, neighborhood children cut across their lot to save walking distance. Given the diagram below, approimate to the nearest foot how many feet of walking distance is saved by cutting across their property instead of walking around the lot ft

9 Section 8. Solving Quadratic Equations by the Quadratic Formula Given the diagram below, approimate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk. 0 ft The hypotenuse of an isosceles right triangle is centimeters longer than either of its legs. Find the eact length of each side. (Hint: An isosceles right triangle is a right triangle whose legs are the same length.) 5. The hypotenuse of an isosceles right triangle is one meter longer than either of its legs. Find the length of each side. 55. Bailey s rectangular dog pen for his Irish setter must have an area of 00 square feet. Also, the length must be 0 feet longer than the width. Find the dimensions of the pen.? 5. An entry in the Peach Festival Poster Contest must be rectangular and have an area of 00 square inches. Furthermore, its length must be 0 inches longer than its width. Find the dimensions each entry must have. 57. A holding pen for cattle must be square and have a diagonal length of 00 meters. a. Find the length of a side of the pen. b. Find the area of the pen. 58. A rectangle is three times longer than it is wide. It has a diagonal of length 50 centimeters. a. Find the dimensions of the rectangle. b. Find the perimeter of the rectangle.? 50 cm 59. The heaviest reported door in the world is the 708. ton radiation shield door in the National Institute for Fusion Science at Toki, Japan. If the height of the door is. feet longer than its width, and its front area (neglecting depth) is 39.9 square feet, find its width and height [Interesting note: the door is. feet thick.] (Source: Guiness World Records) Copyright 0 National Institute for Fusion Science, Japan 0. Christi and Robbie Wegmann are constructing a rectangular stained glass window whose length is 7.3 inches longer than its width. If the area of the window is 59.9 square inches, find its width and length.. The base of a triangle is four more than twice its height. If the area of the triangle is square centimeters, find its base and height.. If a point B divides a line segment such that the smaller portion is to the larger portion as the larger is to the whole, the whole is the length of the golden ratio. A (whole) B The golden ratio was thought by the Greeks to be the most pleasing to the eye, and many of their buildings contained numerous eamples of the golden ratio. The value of the golden ratio is the positive solution of - Find this value. The Wollomombi Falls in Australia have a height of 00 feet. A pebble is thrown upward from the top of the falls with an initial velocity of 0 feet per second. The height of the pebble h after t seconds is given by the equation h -t + 0t Use this equation for Eercises 3 and. 3. How long after the pebble is thrown will it hit the ground? Round to the nearest tenth of a second.. How long after the pebble is thrown will it be 550 feet from the ground? Round to the nearest tenth of a second. A ball is thrown downward from the top of a 80-foot building with an initial velocity of 0 feet per second. The height of the ball h after t seconds is given by the equation h -t - 0t Use this equation to answer Eercises 5 and. 5. How long after the ball is thrown will it strike the ground? Round the result to the nearest tenth of a second. C 50 ft 80 ft

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