Solving Radical Equations
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1 10. Solving Radical Equation Eential Quetion How can you olve an equation that contain quare root? Analyzing a Free-Falling Object MODELING WITH MATHEMATICS To be proficient in math, you need to routinely interpret your mathematical reult in the context of the ituation and reflect on whether the reult make ene. Work with a partner. The table how the time t (in econd) that it take a free-falling object (with no air reitance) to fall d feet. a. Ue the data in the table to ketch the graph of t a a function of d. Ue the coordinate plane below. b. Ue your graph to etimate the time it take the object to fall 0 feet. c. The relationhip between d and t i given by the function d t = 16. Ue thi function to check your etimate in part (b). d. It take 5 econd for the object to hit the ground. How far did it fall? Explain your reaoning. d (feet) t (econd) t d Solving a Square Root Equation Work with a partner. The peed (in feet per econd) of the free-falling object in Exploration 1 i given by the function = 6d. Find the ditance the object ha fallen when it reache each peed. a. = 8 ft/ec b. = 16 ft/ec c. = ft/ec Communicate Your Anwer. How can you olve an equation that contain quare root?. Ue your anwer to Quetion to olve each equation. a. 5 = x + 0 b. = x 18 c. x + = d. = x Section 10. Solving Radical Equation 559
2 10. Leon What You Will Learn Core Vocabulary radical equation, p. 560 Previou radical radical expreion extraneou olution Solve radical equation. Identify extraneou olution. Solve real-life problem involving radical equation. Solving Radical Equation A radical equation i an equation that contain a radical expreion with a variable in the radicand. To olve a radical equation involving a quare root, firt ue propertie of equality to iolate the radical on one ide of the equation. Then ue the following property to eliminate the radical and olve for the variable. Core Concept Squaring Each Side of an Equation Word If two expreion are equal, then their quare are alo equal. Algebra If a = b, then a = b. Solving Radical Equation Solve each equation. a. x + 5 = 1 b. x = 0 x + 5 = =? =? 1 1 = 1 x = 0 9 =? 0 =? 0 0 = 0 a. x + 5 = 1 Write the equation. x = 8 Subtract 5 from each ide. ( x ) = 8 Square each ide of the equation. x = 6 The olution i x = 6. b. x = 0 Write the equation. = x Add x to each ide. = ( x ) Square each ide of the equation. 9 = x The olution i x = 9. Monitoring Progre Help in Englih and Spanih at BigIdeaMath.com Solve the equation. your olution. 1. x = 6. x 7 =. y + 15 =. 1 c = 560 Chapter 10 Radical Function and Equation
3 Solving a Radical Equation x + + = =? =? 19 () + =? = 19 x + + = 19 Original equation x + = 16 Subtract from each ide. x + = Divide each ide by. ( x + ) = Square each ide of the equation. x + = 16 x = 1 Subtract from each ide. The olution i x = y = x + y = x 1 Interection 7 X=5 Y= 1 Solve x 1 = x +. Solving an Equation with Radical on Both Side Method 1 x 1 = x + Write the equation. ( x 1 ) = ( x + ) Square each ide of the equation. x 1 = x + The olution i x = 5. x = 5 Solve for x. Method Graph each ide of the equation, a hown. Ue the interect feature to find the coordinate of the point of interection. The x-value of the point of interection i 5. So, the olution i x = 5. Solving a Radical Equation Involving a Cube Root LOOKING FOR STRUCTURE You can extend the concept taught in Example 1 to olve a radical equation involving a cube root. Intead of quaring each ide of the equation, you cube each ide to eliminate the radical. Solve 5x = 1. 5x = 1 Write the equation. ( 5x ) = 1 Cube each ide of the equation. 5x = 178 x = 6 Solve for x. The olution i x = 6. Monitoring Progre Solve the equation. your olution. Help in Englih and Spanih at BigIdeaMath.com 5. x = = 6 + w 9 7. x + 1 = x 7 8. n = 5n 1 9. y = c + 7 = 10 Section 10. Solving Radical Equation 561
4 ATTEND TO PRECISION To undertand how extraneou olution can be introduced, conider the equation x =. Thi equation ha no real olution, however, you obtain x = after quaring each ide. Identifying Extraneou Solution Squaring each ide of an equation can ometime introduce an extraneou olution. Solve x = x + 6. Identifying an Extraneou Solution x = x + 6 x = ( x + 6 ) x = x + 6 Write the equation. Square each ide of the equation. x x 6 = 0 Subtract x and 6 from each ide. (x )(x + ) = 0 Factor. x = 0 or x + = 0 Zero-Product Property x = or x = Solve for x. STUDY TIP Be ure to alway ubtitute your olution into the original equation to check for extraneou olution. each olution in the original equation. =? + 6 Subtitute for x. =? + 6 =? 9 =? = Becaue x = doe not atify the original equation, it i an extraneou olution. The only olution i x =. Solve 1 + 5n =. Identifying an Extraneou Solution 1 + 5n = 1 + 5(0) =? =? 1 + 5n = Write the equation. 5n = 10 Subtract 1 from each ide. ( 5n ) = ( 10) Square each ide of the equation. 5n = 100 n = 0 Divide each ide by 5. Becaue n = 0 doe not atify the original equation, it i an extraneou olution. So, the equation ha no olution. Monitoring Progre Help in Englih and Spanih at BigIdeaMath.com Solve the equation. your olution(). 11. x = x 1. m + 10 = 1 1. p + 1 = 7p Chapter 10 Radical Function and Equation
5 Solving Real-Life Problem Modeling with Mathematic STUDY TIP The period of a pendulum i the amount of time it take for the pendulum to wing back and forth. The period P (in econd) of a pendulum i given by the L function P = π, where L i the pendulum length (in feet). A pendulum ha a period of econd. I thi pendulum twice a long a a pendulum with a period of econd? Explain your reaoning. 1. Undertand the Problem You are given a function that repreent the period P of a pendulum baed on it length L. You need to find and compare the value of L for two value of P.. Make a Plan Subtitute P = and P = into the function and olve for L. Then compare the value.. Solve the Problem P = π L Write the function. P = π L = π L π = L 1 π = L 1 π = L Subtitute for P. = π L Divide each ide by π. Square each ide and implify. π = L π = L π = L π = L Multiply each ide by. 18 π = L. L Ue a calculator L No, the length of the pendulum with a period of econd i 18 π = time longer than the length of a pendulum with a π period of econd.. Look Back Ue the trace feature of a graphing calculator to check your olution. 6 6 y = π x y = π x 0 X=. 0 Y= X= Y= Monitoring Progre Help in Englih and Spanih at BigIdeaMath.com 1. What i the length of a pendulum that ha a period of.5 econd? Section 10. Solving Radical Equation 56
6 10. Exercie Dynamic Solution available at BigIdeaMath.com Vocabulary and Core Concept 1. VOCABULARY Why hould you check every olution of a radical equation?. WHICH ONE DOESN T BELONG? Which equation doe not belong with the other three? Explain your reaoning. x + 6 = 10 x + = x 5 = x 1 = 16 Monitoring Progre and Modeling with Mathematic In Exercie 1, olve the equation. your olution. (See Example 1.). x = 9. y = 5. 7 = m 5 6. p 7 = 1 7. c + 1 = 8. x + 6 = 8 9. a = = 7 r 11. y 18 = 1. q + 5 = 11 In Exercie 1 0, olve the equation. your olution. (See Example.) 1. a + 5 = 9 1. b = 15. x + = y = = 5r = p 9 = c 11 = MODELING WITH MATHEMATICS The Cave of Swallow i a natural open-air pit cave in the tate of San Lui Potoí, Mexico. The 10-footdeep cave wa a popular detination for BASE jumper. The function t = 1 d repreent the time t (in econd) that it take a BASE jumper to fall d feet. How far doe a BASE jumper fall in econd?. MODELING WITH MATHEMATICS The edge length of a cube with a urface area of A i given by = A. What i the urface area 6 of a cube with an edge length of inche? In Exercie 6, ue the graph to olve the equation.. x + = x +. x + 1 = x y y = x + x y = x + 6 y y = x + 1 y = x 6 x 5. x + x = 0 6. x + 5 x + 7 = 0 y y = x + x y = x y = x + 5 y = x + 7 In Exercie 7, olve the equation. your olution. (See Example.) 7. x 9 = x 8. y + 1 = y 8 9. g + 1 = 7g h 7 = 6h p = p 8. v 5 = v + 5. c + 1 c = 0. 5r 8r = 0 y 1 x 56 Chapter 10 Radical Function and Equation
7 MATHEMATICAL CONNECTIONS In Exercie 5 and 6, find the value of x. 5. Perimeter = cm 6. Area = 5x ft 6x 5 cm x + 1 ft 60. x = 1 x x = 1 x x + x 1 = 0 (x )(x + 6) = 0 x = or x = 6 The olution are x = and x = 6. cm ft In Exercie 7, olve the equation. your olution. (See Example.) 7. x = = 8g y = r + 19 = + 9 =. 5 = 10x + 15 y + 6 = 5y. 7j = j + In Exercie 5 8, determine which olution, if any, i an extraneou olution. 5. 6x 5 = x; x = 5, x = REASONING Explain how to ue mental math to olve x + 5 = WRITING Explain how you would olve m + m = MODELING WITH MATHEMATICS The formula V = PR relate the voltage V (in volt), power P (in watt), and reitance R (in ohm) of an electrical circuit. The hair dryer hown i on a 10-volt circuit. I the reitance of the hair dryer half a much a the reitance of the ame hair dryer on a 0-volt circuit? Explain your reaoning. (See Example 7.) 6. y + = y; y = 1, y = 7. 1p + 16 = p; p = 1, p = 8. g = 18 7g ; g =, g = 1 In Exercie 9 58, olve the equation. your olution(). (See Example 5 and 6.) 9. y = 5y x = x a = a 5. q = 10q p = 5. n 11 = m + = b 8 = r + = r = 1 ERROR ANALYSIS In Exercie 59 and 60, decribe and correct the error in olving the equation x = 1 5 x = 10 5x = 100 x = 0 6. MODELING WITH MATHEMATICS The time t (in econd) it take a trapeze artit to wing back and forth i repreented by the function t = π r, where r i the rope length (in feet). It take the trapeze artit 6 econd to wing back and forth. I thi rope a long a the rope ued when it take the trapeze artit econd to wing back and forth? Explain your reaoning. REASONING In Exercie 65 68, determine whether the tatement i true or fale. If it i fale, explain why. 65. If a = b, then ( a ) = b. 66. If a = b, then a = b. 67. If a = b, then a = b. 68. If a = b, then a = ( b ). Section 10. Solving Radical Equation 565
8 69. COMPARING METHODS Conider the equation x + = x. a. Solve the equation by graphing. Decribe the proce. b. Solve the equation algebraically. Decribe the proce. c. Which method do you prefer? Explain your reaoning. USING STRUCTURE In Exercie 7 78, olve the equation. your olution. 7. m + 15 = m x + 1 = x y y = c 8 c = h h = HOW DO YOU SEE IT? The graph how two radical function. y y = x + y = x x a. Write an equation whoe olution i the x-coordinate of the point of interection of the graph. b. Ue the graph to olve the equation. 71. MATHEMATICAL CONNECTIONS The lant height of a cone with a radiu of r and a height of h i given by = r + h. The lant height of the two cone are equal. Find the radiu of each cone. r 7. CRITICAL THINKING How i quaring x + different from quaring x +? Maintaining Mathematical Proficiency Find the product. (Section 7.) r z + z = OPEN-ENDED Write a radical equation that ha a olution of x = OPEN-ENDED Write a radical equation that ha x = and x = a olution. 81. MAKING AN ARGUMENT Your friend ay the equation (x + 5) = x + 5 i alway true, becaue after implifying the left ide of the equation, the reult i an equation with infinitely many olution. I your friend correct? Explain. 8. THOUGHT PROVOKING Solve the equation x + 1 = x. Show your work and explain your tep. 8. MODELING WITH MATHEMATICS The frequency f (in cycle per econd) of a tring of an electric guitar i given by the equation f = 1 T, where i the length of the m tring (in meter), T i the tring tenion (in newton), and m i the tring ma per unit length (in kilogram per meter). The high E tring of an electric guitar i 0.6 meter long with a ma per unit length of kilogram per meter. a. How much tenion i required to produce a frequency of about 0 cycle per econd? b. Would you need more or le tenion to create the ame frequency on a tring with greater ma per unit length? Explain. 8. (x + 8)(x ) 85. (p 1)(p + 5) 86. ( + )( + ) Graph the function. Compare the graph to the graph of f (x) = x. (Section 8.1) Reviewing what you learned in previou grade and leon 87. r(x) = x 88. g(x) = x 89. h(x) = 5x 566 Chapter 10 Radical Function and Equation
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