8.2 Solving Quadratic Equations by the Quadratic Formula


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1 Section 8. Solving Quadratic Equations by the Quadratic Formula Solving Quadratic Equations by the Quadratic Formula S Solve Quadratic Equations by Using the Quadratic Formula. Determine the Number and Type of Solutions of a Quadratic Equation by Using the Discriminant. 3 Solve Problems Modeled by Quadratic Equations. Solving Quadratic Equations by Using the Quadratic Formula Any quadratic equation can be solved by completing the square. Since the same sequence of steps is repeated each time we complete the square, let s complete the square for a general quadratic equation, a + b + c 0, a 0. By doing so, we find a pattern for the solutions of a quadratic equation known as the quadratic formula. Recall that to complete the square for an equation such as a + b + c 0, we first divide both sides by the coefficient of. a + b + c 0 + b a + c a 0 Divide both sides by a, the coefficient of. + b a  c a Subtract the constant c from both sides. a Net, find the square of half b, the coefficient of. a a b a b b a and a b a b b a Add this result to both sides of the equation. + b a + b a  c a + b a + b a + b a c # a a # a + b a + b a b  ac a a + b a b b  ac a + b a + b a { b  ac B a + b a { b  ac a  b a { b  ac a b { b  ac a Add b to both sides. a Find a common denominator on the right side. Simplify the right side. Factor the perfect square trinomial on the left side. Apply the square root property. Simplify the radical. Subtract b from both sides. a Simplify. This equation identifies the solutions of the general quadratic equation in standard form and is called the quadratic formula. It can be used to solve any equation written in standard form a + b + c 0 as long as a is not 0. Quadratic Formula A quadratic equation written in the form a + b + c 0 has the solutions b { b  ac a
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: 38. 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
3 Section 8. Solving Quadratic Equations by the Quadratic Formula 87 CONCEPT CHECK For the quadratic equation 7, which substitution is correct? a. a, b 0, and c 77 b. a, b 0, and c 7 c. a 0, b 0, and c 7 d. a, b, and c 77 EXAMPLE 3 Solve: m  m + 0. Solution We could use the quadratic formula with a, b , and c. Instead, we find a simpler, equivalent standard form equation whose coefficients are not fractions. Multiply both sides of the equation by the LCD to clear fractions. a m  m + b # 0 m  m + 0 Simplify. Substitute a, b , and c into the quadratic formula and simplify. m   {   # {  8 The solutions are + and . 3 Solve: { 8 { { { EXAMPLE Solve: Solution The equation in standard form is Thus, let a 3, b, and c 3 in the quadratic formula.  {  33 # 3  { { { i35 The solutions are  + i35 and   i35. Solve:  . CONCEPT CHECK What is the first step in solving using the quadratic formula? Answer to Concept Checks: a Write the equation in standard form. In Eample, the equation had real roots,  and 5.In Eample, 3 the equation (written in standard form) had no real roots. How do their related graphs compare? Recall that the intercepts of f occur
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 { 000 { 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 850 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 00foot 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
7 Section 8. Solving Quadratic Equations by the Quadratic Formula 9. INTERPRET. Check: We check our calculations from the quadratic formula. Since the time won t be negative, we reject the proposed solution State: The time it takes for the object to strike the ground is eactly seconds 3.9 seconds. 7 A toy rocket is shot upward from the top of a building, 5 feet high, with an initial velocity of 0 feet per second. The height h in feet of the rocket after t seconds is h t + 0t + 5 How long after the rocket is launched will it strike the ground? Round to the nearest tenth of a second. Vocabulary, Readiness & Video Check Fill in each blank.. The quadratic formula is.. For + + 0, if a, then b and c. 3. For , if a 5, then b and c.. For 70, if a 7, then b and c. 5. For + 9 0, if c 9, then a and b.. The correct simplified form of 5 { 0 is. 5 a. { 0 b. c. { d. {5 MartinGay Interactive Videos See Video 8. Watch the section lecture video and answer the following questions Based on Eamples 3, answer the following. a. Must a quadratic equation be written in standard form in order to use the quadratic formula? Why or why not? b. Must fractions be cleared from an equation before using the quadratic formula? Why or why not? 8. Based on Eample and the lecture before, complete the following statements. The discriminant is the in the quadratic formula and can be used to find the number and type of solutions of a quadratic equation without the equation. To use the discriminant, the quadratic equation needs to be written in form. 9. In Eample 5, the value of is found, which is then used to find the dimensions of the triangle. Yet all this work still does solve the problem. Eplain.
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 80foot 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
10 9 CHAPTER 8 Quadratic Equations and Functions. How long after the ball is thrown will it be 50 feet from the ground? Round the result to the nearest tenth of a second. REVIEW AND PREVIEW Solve each equation. See Sections. and y Factor. See Section y + y z  3z CONCEPT EXTENSIONS z 5 z  3 For each quadratic equation, choose the correct substitution for a, b, and c in the standard form a + b + c a. a, b 0, c 0 b. a, b 0, c 0 c. a 0, b, c 0 d. a, b, c a. a, b 5, c  b. a, b , c 5 c. a, b 5, c d. a, b, c Solve Eercise by factoring. Eplain the result. 78. Solve Eercise by factoring. Eplain the result. Use the quadratic formula and a calculator to approimate each solution to the nearest tenth The accompanying graph shows the daily low temperatures for one week in New Orleans, Louisiana. Degrees Fahrenheit Sun. Mon. Tues. Wed. Thu. Fri. Sat. 8. Between which days of the week was there the greatest decrease in the low temperature? 8. Between which days of the week was there the greatest increase in the low temperature? 83. Which day of the week had the lowest low temperature? 8. Use the graph to estimate the low temperature on Thursday. Notice that the shape of the temperature graph is similar to the curve drawn. In fact, this graph can be modeled by the quadratic function f , where f() is the temperature in degrees Fahrenheit and is the number of days from Sunday. (This graph is shown in blue.) Use this function to answer Eercises 85 and Use the quadratic function given to approimate the temperature on Thursday. Does your answer agree with the graph? 8. Use the function given and the quadratic formula to find when the temperature was 35 F. [Hint: Let f 35 and solve for.] Round your answer to one decimal place and interpret your result. Does your answer agree with the graph? 87. The number of college students in the United States can be modeled by the quadratic function f ,8, where f is the number of college students in thousands of students, and is the number of years after 000. (Source: Based on data from the U.S. Department of Education) a. Find the number of college students in the United States in 00. b. If the trend described by this model continues, find the year after 000 in which the population of American college students reaches,500 students. 88. The projected number of WiFienabled cell phones in the United States can be modeled by the quadratic function c , where c() is the projected number of WiFienabled cell phones in millions and is the number of years after 009. Round to the nearest million. (Source: Techcrunchies.com) a. Find the number of WiFienabled cell phones in the United States in 00. b. Find the projected number of WiFienabled cell phones in the United States in 0. c. If the trend described by this model continues, find the year in which the projected number of WiFienabled cell phones in the United States reaches 50 million. 89. The average total daily supply y of motor gasoline (in thousands of barrels per day) in the United States for the period can be approimated by the equation y , where is the number of years after 000. (Source: Based on data from the Energy Information Administration) a. Find the average total daily supply of motor gasoline in 00. b. According to this model, in what year, from 000 to 008, was the average total daily supply of gasoline 935 thousand barrels per day? c. According to this model, in what year, from 009 on, will the average total supply of gasoline be 935 thousand barrels per day?
11 Section 8.3 Solving Equations by Using Quadratic Methods The relationship between body weight and the Recommended Dietary Allowance (RDA) for vitamin A in children up to age 0 is modeled by the quadratic equation y , where y is the RDA for vitamin A in micrograms for a child whose weight is pounds. (Source: Based on data from the Food and Nutrition Board, National Academy of Sciences Institute of Medicine, 989) a. Determine the vitamin A requirements of a child who weighs 35 pounds. b. What is the weight of a child whose RDA of vitamin A is 00 micrograms? Round your answer to the nearest pound. The solutions of the quadratic equation a + b + c 0 are b + b  ac a and b  b  ac. a 9. Show that the sum of these solutions is b a. 9. Show that the product of these solutions is c a. Use the quadratic formula to solve each quadratic equation (Hint: a 3, b , c ) Use a graphing calculator to solve Eercises 3 and Use a graphing calculator to solve Eercises and. Recall that the discriminant also tells us the number of intercepts of the related function. 0. Check the results of Eercise 9 by graphing y Check the results of Eercise 50 by graphing y Solving Equations by Using Quadratic Methods S Solve Various Equations That Are Quadratic in Form. Solve Problems That Lead to Quadratic Equations. Solving Equations That Are Quadratic in Form In this section, we discuss various types of equations that can be solved in part by using the methods for solving quadratic equations. Once each equation is simplified, you may want to use these steps when deciding which method to use to solve the quadratic equation. Solving a Quadratic Equation Step. If the equation is in the form a + b c, use the square root property and solve. If not, go to Step. Step. Write the equation in standard form: a + b + c 0. Step 3. Try to solve the equation by the factoring method. If not possible, go to Step. Step. Solve the equation by the quadratic formula. The first eample is a radical equation that becomes a quadratic equation once we square both sides. EXAMPLE Solve: Solution Recall that to solve a radical equation, first get the radical alone on one side of the equation. Then square both sides. (Continued on net page) or Add to both sides. Square both sides. Set the equation equal to 0.
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