Use Square Roots to Solve Quadratic Equations


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1 10.4 Use Square Roots to Solve Quadratic Equations Before You solved a quadratic equation by graphing. Now You will solve a quadratic equation by finding square roots. Why? So you can solve a problem about a falling object, as in Example 5. FPO Key Vocabulary square root, p. 110 perfect square, p. 111 To use square roots to solve a quadratic equation of the form ax 2 1 c 5 0, first isolate x 2 on one side to obtain x 2 5 d. Then use the following information about the solutions of x 2 5 d to solve the equation. KEY CONCEPT For Your Notebook READING Recall that in this course, solutions refers to realnumber solutions. Solving x 2 5 d by Taking Square Roots If d. 0, then x 2 5 d has two solutions: x 56Ï } d. If d 5 0, then x 2 5 d has one solution: x 5 0. If d, 0, then x 2 5 d has no solution. d > 0 d 5 0 d <0 y x E XAMPLE 1 Solve quadratic equations Solve the equation. a. 2x b. m c. b ANOTHER WAY You can also use factoring to solve 2x : 2x (x 2 2 4) 5 0 2(x 2 2)(x 1 2) 5 0 x 5 2 or x 522 Solution a. 2x Write original equation. x Divide each side by 2. x 56Ï } Take square roots of each side. Simplify. c The solutions are 22 and 2. b. m Write original equation. m Add 18 to each side. m 5 0 The square root of 0 is 0. c The solution is 0. c. b Write original equation. b Subtract 12 from each side. c Negative real numbers do not have real square roots. So, there is no solution. 652 Chapter 10 Quadratic Equations and Functions
2 SIMPLIFYING SQUARE ROOTS In cases where you need to take the square root of a fraction whose numerator and denominator are perfect squares, the radical can be written as a fraction. For example, Î } 16 } can be written 25 as } 4 because 5 1 } } E XAMPLE 2 Take square roots of a fraction Solve 4z Solution 4z Write original equation. z } 4 Divide each side by 4. z 56Î } 9 }4 z 56 3 } 2 Take square roots of each side. Simplify. c The solutions are 2 3 } 2 and 3 } 2. APPROXIMATING SQUARE ROOTS In cases where d in the equation x 2 5 d is not a perfect square or a fraction whose numerator and denominator are not perfect squares, you need to approximate the square root. A calculator can be used to find an approximation. E XAMPLE 3 Approximate solutions of a quadratic equation Solve 3x Round the solutions to the nearest hundredth. Solution 3x x Write original equation. Add 11 to each side. x Divide each side by 3. x 56Ï } 6 Take square roots of each side. x ø Use a calculator. Round to the nearest hundredth. c The solutions are about and about GUIDED PRACTICE for Examples 1, 2, and 3 Solve the equation. 1. c w x x m b Solve the equation. Round the solutions to the nearest hundredth. 7. x k p Use Square Roots to Solve Quadratic Equations 653
3 E XAMPLE 4 Solve a quadratic equation Solve 6(x 2 4) Round the solutions to the nearest hundredth. 6(x 2 4) Write original equation. (x 2 4) Divide each side by 6. x Ï } 7 x Ï } 7 Take square roots of each side. Add 4 to each side. c The solutions are 4 1 Ï } 7 ø 6.65 and 4 2 Ï } 7 ø CHECK To check the solutions, first write the equation so that 0 is on one side as follows: 6(x 2 4) Then graph the related function y 5 6(x 2 4) The xintercepts appear to be about 6.6 and about 1.3. So, each solution checks E XAMPLE 5 Solve a multistep problem ANOTHER WAY For alternative methods for solving the problem in Example 5, turn to page 659 for the Problem Solving Workshop. SPORTS EVENT During an ice hockey game, a remotecontrolled blimp flies above the crowd and drops a numbered tabletennis ball. The number on the ball corresponds to a prize. Use the information in the diagram to find the amount of time that the ball is in the air. Solution DETERMINE VELOCITY When an object is dropped, it has an initial vertical velocity of 0 feet per second. STEP 1 Use the vertical motion model to write an equation for the height h (in feet) of the ball as a function of time t (in seconds). h 5216t 2 1 vt 1 s Vertical motion model h 5216t 2 1 0t 1 45 Substitute for v and s. STEP 2 Find the amount of time the ball is in the air by substituting 17 for h and solving for t. h 5216t Write model. 45 ft 17 ft Not drawn to scale t Substitute 17 for h t 2 Subtract 45 from each side. 28 } 16 5 t 2 Divide each side by 216. INTERPRET SOLUTION Because the time cannot be a negative number, ignore the negative square root. Î } 28 } 16 5 t Take positive square root ø t Use a calculator. c The ball is in the air for about 1.32 seconds. 654 Chapter 10 Quadratic Equations and Functions
4 GUIDED PRACTICE for Examples 4 and 5 Solve the equation. Round the solutions to the nearest hundredth, if necessary (x 2 2) (q 2 3) (t 1 5) WHAT IF? In Example 5, suppose the tabletennis ball is released 58 feet above the ground and is caught 12 feet above the ground. Find the amount of time that the ball is in the air. Round your answer to the nearest hundredth of a second EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKEDOUT SOLUTIONS on p. WS1 for Exs. 25 and 59 5 STANDARDIZED TEST PRACTICE Exs. 2, 15, 16, 29, 51, 52, 57, and 60 5 MULTIPLE REPRESENTATIONS Ex VOCABULARY Copy and complete: If b 2 5 a, then b is a(n)? of a. 2. WRITING Describe two methods for solving a quadratic equation of the form ax 2 1 c 5 0. EXAMPLES 1 and 2 on pp for Exs SOLVING EQUATIONS Solve the equation. 3. 3x x x m d a g w q b z n MULTIPLE CHOICE Which of the following is a solution of the equation n ? A 5 B 10 C 25 D MULTIPLE CHOICE Which of the following is a solution of the equation x ? A 2 6 } 5 B 1 } 6 C 5 } 6 D 5 EXAMPLE 3 on p. 653 for Exs APPROXIMATING SQUARE ROOTS Solve the equation. Round the solutions to the nearest hundredth. 17. x x x a k p m z c d b n MULTIPLE CHOICE The equation 17 2 } 1 4 x has a solution between which two integers? A 1 and 2 B 2 and 3 C 3 and 4 D 4 and Use Square Roots to Solve Quadratic Equations 655
5 ERROR ANALYSIS Describe and correct the error in solving the equation x d x x x x 5 Ï } 36 x 5 6 The solution is 6. 7d d d 2 52} 11 7 d ø The solutions are about and about EXAMPLE 4 on p. 654 for Exs SOLVING EQUATIONS Solve the equation. Round the solutions to the nearest hundredth. 32. (x 2 7) (x 2 3) (x 1 4) (m 1 5) (a 2 2) (z 1 14) } 2 ( c 2 8) } 2 (n 1 1) } 3 (k 2 6) SOLVING EQUATIONS Solve the equation. Round the solutions to the nearest hundredth, if necessary x x (x 2 1 5) x (4x 2 2 3) t 2 5 } w 2 7 } (4m 2 2 6) GEOMETRY Use the given area A of the circle to find the radius r or the diameter d to the nearest hundredth. 47. A 5 144π in A 5 21π m A 5 34π ft 2 r r d 50. REASONING An equation of the graph shown is y 5 } 1 (x 2 2) Two points on the parabola have 2 ycoordinates of 9. Find the xcoordinates of these points. 1 y 1 x 51. SHORT RESPONSE Solve x without using a calculator. Explain your reasoning. 52. OPEN ENDED Give values for a and c so that ax 2 1 c 5 0 has (a) two solutions, (b) one solution, and (c) no solution. CHALLENGE Solve the equation without graphing. 53. x x x x x x WORKEDOUT SOLUTIONS on p. WS1 5 STANDARDIZED TEST PRACTICE
6 PROBLEM SOLVING EXAMPLE 5 on p. 654 for Exs FALLING OBJECT Fenway Park is a Major League Baseball park in Boston, Massachusetts. The park offers seats on top of the left field wall. A person sitting in one of these seats accidentally drops his sunglasses on the field. The height h (in feet) of the sunglasses can be modeled by the function h 5216t where t is the time (in seconds) since the sunglasses were dropped. Find the time it takes for the sunglasses to reach the field. Round your answer to the nearest hundredth of a second. 57. MULTIPLE CHOICE Which equation can be used to find the time it takes for an object to hit the ground after it was dropped from a height of 68 feet? A 216t B 216t C 216t D 216t INTERNET USAGE For the period , the number y (in thousands) of Internet users worldwide can be modeled by the function y 5 12,697x ,722 where x is the number of years since Between which two years did the number of Internet users worldwide reach 100,000,000? 59. GEMOLOGY To find the weight w (in carats) of round faceted gems, gemologists use the formula w D 2 ds where D is the diameter (in millimeters) of the gem, d is the depth (in millimeters) of the gem, and s is the specific gravity of the gem. Find the diameter to the nearest tenth of a millimeter of each round faceted gem in the table. Gem Weight (carats) Depth (mm) Specific gravity Diameter (mm) a. b. c. Amethyst ? Diamond ? Ruby ? 60. SHORT RESPONSE In deep water, the speed s (in meters per second) of a series of waves and the wavelength L (in meters) of the waves are related by the equation 2πs L. L Crest Crest The wavelength L is the distance between one crest and the next. a. Find the speed to the nearest hundredth of a meter per second of a series of waves with the following wavelengths: 6 meters, 10 meters, and 25 meters. (Use 3.14 for π.) b. Does the speed of a series of waves increase or decrease as the wavelength of the waves increases? Explain Use Square Roots to Solve Quadratic Equations 657
7 61. MULTISTEP PROBLEM The Doyle log rule is a formula used to estimate the amount of lumber that can be sawn from logs of various sizes. The amount of lumber L(D 2 4)2 V (in board feet) is given by V 5} where L is 16 the length (in feet) of a log and D is the smallend diameter (in inches) of the log. a. Solve the formula for D. b. Use the rewritten formula to find the diameters, to the nearest tenth of a foot, of logs that will yield 50 board feet and have the following lengths: 16 feet, 18 feet, 20 feet, and 22 feet. Diameter Boards 62. MULTIPLE REPRESENTATIONS A ride at an amusement park lifts seated riders 250 feet above the ground. Then the riders are dropped. They experience free fall until the brakes are activated at 105 feet above the ground. a. Writing an Equation Use the vertical motion model to write an equation for the height h (in feet) of the riders as a function of the time t (in seconds) into the free fall. b. Making a Table Make a table that shows the height of the riders after 0, 1, 2, 3, and 4 seconds. Use the table to estimate the amount of time the riders experience free fall. c. Solving an Equation Use the equation to find the amount of time, to the nearest tenth of a second, that the riders experience free fall. 63. CHALLENGE The height h (in feet) of a dropped object on any planet can be modeled by h 5 2} g t 2 1 s where g is the acceleration (in feet per 2 second per second) due to the planet s gravity, t is the time (in seconds) after the object is dropped, and s is the initial height (in feet) of the object. Suppose the same object is dropped from the same height on Earth and Mars. Given that g is 32 feet per second per second on Earth and 12 feet per second per second on Mars, on which planet will the object hit the ground first? Explain. MIXED REVIEW PREVIEW Prepare for Lesson 10.5 in Exs Evaluate the power. (p. 2) } } } } Write an equation of the line with the given slope and yintercept. (p. 283) 68. slope: slope: slope: 3 yintercept: 11 yintercept: 27 yintercept: 22 Write an equation of the line that passes through the given point and is perpendicular to the given line. (p. 319) 71. (1, 21), y 5 2x 72. (0, 8), y 5 4x (29, 24), y 523x EXTRA PRACTICE for Lesson 10.4, p. 947 ONLINE QUIZ at classzone.com
8 LESSON 10.4 Using ALTERNATIVE METHODS Another Way to Solve Example 5, page 654 MULTIPLE REPRESENTATIONS In Example 5 on page 654, you saw how to solve a problem about a dropped tabletennis ball by using a square root. You can also solve the problem by using factoring or by using a table. P ROBLEM SPORTS EVENT During an ice hockey game, a remotecontrolled blimp flies above the crowd and drops a numbered tabletennis ball. The number on the ball corresponds to a prize. Use the information in the diagram to find the amount of time that the ball is in the air. 45 ft 17 ft Not drawn to scale M ETHOD 1 Using Factoring One alternative approach is to use factoring. STEP 1 Write an equation for the height h (in feet) of the ball as a function of time t (in seconds) after it is dropped using the vertical motion model. h 5216t 2 1 vt 1 s Vertical motion model h 5216t 2 1 0t 1 45 Substitute 0 for v and 45 for s. STEP 2 Substitute 17 for h to find the time it takes the ball to reach a height of 17 feet. Then write the equation so that 0 is on one side t Substitute 17 for h t Subtract 17 from each side. USE AN APPROXIMATION By replacing 28 with 25, you will obtain an answer that is an approximation of the amount of time that the ball is in the air. STEP 3 Solve the equation by factoring. Replace 28 with the closest perfect square, 25, so that the right side of the equation is factorable as a difference of two squares t Use 25 as an approximation for (16t ) Factor out (4t 2 5)(4t 1 5) Difference of two squares pattern 4t or 4t Zeroproduct property t 5 5 } 4 or t 52 5 } 4 Solve for t. c The ball is in the air about 5 } 4, or 1.25, seconds. Using Alternative Methods 659
9 M ETHOD 2 Using a Table Another approach is to make and use a table. STEP 1 Make a table that shows the height h (in feet) of the ball by substituting values for time t (in seconds) in the function h 5216t Use increments of 1 second. STEP 2 Identify the time interval in which the height of the ball is 17 feet. This happens between 1 and 2 seconds. STEP 3 Make a second table using increments of 0.1 second to get a closer approximation. c The ball is in the air about 1.3 seconds. Time t (seconds) Height h (feet) Time t (seconds) Height h (feet) P RACTICE 1. WHAT IF? In the problem on page 659, suppose the ball is caught at a height of 10 feet. For how many seconds is the ball in the air? Solve this problem using two different methods. 2. OPENENDED Describe a problem about a dropped object. Then solve the problem and explain what your solution means in this situation. 3. GEOMETRY The box below is a rectangular prism with the dimensions shown. 5x in. 5 in. x in. a. Write an equation that gives the volume V (in cubic inches) of the box as a function of x. b. The volume of the box is 83 cubic inches. Find the dimensions of the box. Use factoring to solve the problem. c. Make a table to check your answer from part (b). 4. TRAPEZE You are learning how to perform on a trapeze. While hanging from a still trapeze bar, your shoe comes loose and falls to a safety net that is 6 feet off the ground. If your shoe falls from a height of 54 feet, how long does it take your shoe to hit the net? Choose any method for solving the problem. Show your steps. 5. ERROR ANALYSIS A student solved the problem in Exercise 4 as shown below. Describe and correct the error. Let t be the time (in seconds) that the shoe is in the air t t Replace 60 with the closest perfect square, t (t 2 2)(t 1 2) t 5 2 or t 522 It takes about 2 seconds. 660 Chapter 10 Quadratic Equations and Functions
10 MIXED REVIEW of Problem Solving STATE TEST PRACTICE classzone.com Lessons MULTISTEP PROBLEM A company s yearly profits from 1996 to 2006 can be modeled by the function y 5 x 2 2 8x 1 80 where y is the profit (in thousands of dollars) and x is the number of years since a. In what year did the company experience its lowest yearly profit? b. What was the lowest yearly profit? 2. MULTISTEP PROBLEM Use the rectangle below. 2x ft (14 2 x) ft a. Find the value of x that gives the greatest possible area of the rectangle. b. What is the greatest possible area of the rectangle? 3. EXTENDED RESPONSE You throw a lacrosse ball twice using a lacrosse stick. 4. OPENENDED Describe a realworld situation of an object being dropped. Then write an equation that models the height of the object as a function of time. Use the equation to determine the time it takes the object to hit the ground. 5. SHORT RESPONSE A football player is attempting a field goal. The path of the kicked football can be modeled by the graph of y x x where x is the horizontal distance (in yards) traveled by the football and y is the corresponding height (in feet) of the football. Will the football pass over the goal post that is 10 feet above the ground and 45 yards away? Explain. 6. GRIDDED ANSWER The force F (in newtons) a rider feels while a train goes around a curve is given by F 5 } mv2 r where m is the mass (in kilograms) of the rider, v is the velocity (in meters per second) of the train, and r is the radius (in meters) of the curve. A rider with a mass of 75 kilograms experiences a force of 18,150 newtons, while going around a curve that has a radius of 8 meters. Find the velocity (in meters per second) the train travels around the curve. 7. SHORT RESPONSE The opening of the tunnel shown can be modeled by the graph of the equation y x x 2 12 where x and y are measured in feet. a. For your first throw, the ball is released 8 feet above the ground with an initial vertical velocity of 35 feet per second. Use the vertical motion model to write an equation for the height h (in feet) of the ball as a function of time t (in seconds). b. For your second throw, the ball is released 7 feet above the ground with an initial vertical velocity of 45 feet per second. Use the vertical motion model to write an equation for the height h (in feet) of the ball as a function of time t (in seconds). c. If no one catches either throw, for which throw is the ball in the air longer? Explain. a. Find the maximum height of the tunnel. b. A semi trailer is 7.5 feet wide, and the top of the trailer is 10.5 feet above the ground. Given that traffic travels one way on one lane through the center of the tunnel, will the semi trailer fit through the opening of the tunnel? Explain. Mixed Review of Problem Solving 661
11 Investigating Algebra ACTIVITY Use before Lesson 10.5 Algebra classzone.com 10.5 Completing the Square Using Algebra Tiles MATERIALS algebra tiles QUESTION How can you use algebra tiles to complete the square? For an expression of the form x 2 1 bx, you can add a constant c to the expression so that the expression x 2 1 bx 1 c is a perfect square trinomial. This process is called completing the square. E XPLORE Complete the square Find the value of c that makes x 2 1 4x 1 c a perfect square trinomial. STEP 1 Model expression Use algebra tiles to model the expression x 2 1 4x. You will need one x 2 tile and four xtiles for this expression. STEP 2 Rearrange tiles Arrange the tiles to form a square. The arrangement will be incomplete in one of the corners. STEP 3 Complete the square Determine the number of 1tiles needed to complete the square. The number of 1tiles is the value of c. So, the perfect square trinomial is x 2 1 4x 1 4 or (x 1 2) 2. DRAW CONCLUSIONS Use your observations to complete these exercises 1. Copy and complete the table using algebra tiles. Expression Number of 1tiles needed to complete the square Expression written as a square x 2 1 4x 4 x 2 1 4x (x 1 2) 2 x 2 1 6x?? x 2 1 8x?? x x?? 2. In the statement x 2 1 bx 1 c 5 (x 1 d) 2, how are b and d related? How are c and d related? 3. Use your answer to Exercise 2 to predict the number of 1tiles you would need to add to complete the square for the expression x x. 662 Chapter 10 Quadratic Equations and Functions
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