The Quadratic Formula

LESSON 7.4 EXAMPLE A Solution The Qudrtic Formul Although you cn lwys use grph of qudrtic function to pproximte the x-intercepts, you re often not ble to find exct solutions. This lesson will develop procedure to find the exct roots of qudrtic eqution. Consider gin this sitution from Exmple C in the previous lesson. Nor hits softbll stright up t speed of 10 ft/s. Her bt contcts the bll t height of 3 ft bove the ground. Recll tht the eqution relting height in feet, y, nd time in seconds, x, is y 16x 10x 3. How long will it be until the bll hits the ground? The height will be zero when the bll hits the ground, so you wnt to find the solutions to the eqution 16x 10x 3 0. You cn pproximte the x-intercepts by grphing, but you my not be ble to find the exct x-intercept. This vlue represents 5.59908 10 10, or 0.000000000559908..., which is very close to zero. You will not be ble to fctor this eqution using rectngle digrm, so you cn t use the zero-product property. Insted, to solve this eqution symboliclly, first write the eqution in the form (x h) k 0. 16x 10x 3 0 Originl eqution. 16x 10x 3 Subtrct the constnt from both sides. 16x 7.5x? 3 Fctor to get the leding coefficient 1. 16 x 7.5x 3.75 3 163.75 Complete the squre. 16x 3.75 8 Fctor nd combine like terms. x 3.75 14.5 Divide by 16. x 3.75 14.5 Tke the squre root of both sides. x 3.75 14.5 Add 3.75 to both sides. x 3.75 14.5 or x 3.75 14.5 Write the two exct solutions to the eqution. x 7.55 or x 0.05 Approximte the vlues of x. The zeros of the function re x 7.55 nd x 0.05. The negtive time, 0.05 s, does not mke sense in this sitution, so the bll hits the ground fter pproximtely 7.55 s. OBJECTIVES Use the vertex form of qudrtic eqution to find the eqution s roots Derive the qudrtic formul by completing the squre Use the qudrtic formul to solve ppliction problems OUTLINE One dy: 15 min Exmple A 10 min Investigtion 10 min Discuss Investigtion 5 min Exmple B 5 min Exercises MATERIALS LESSON 7.4 Investigtion Worksheet, optionl Clcultor Note 7D, optionl Sketchpd demonstrtion Qudrtic Functions, optionl ADDITIONAL SUPPORT Lesson 7.4 More Prctice Your Skills Lesson 7.4 Condensed Lessons (in English or Spnish) TestCheck worksheets TEACHING THE LESSON This lesson derives the qudrtic formul. Students who used Discovering Algebr my recll it from Chpter 9 of tht book. DIFFERENTIATING INSTRUCTION ELL Students should be ble to differentite mong qudrtic functions (the reltion ship between two vribles), qudrtic equtions ( specific vlue of the func tions), nd the qudrtic formul ( method used to solve qudrtic equtions). Hve students provide verbl nd symbolic exmples of ech. LESSON OBJECTIVES Extr Support Crete histogrm nd Advnced stem-nd-lef plot of dt set Students should Given work list through of dt, use Ask clcultor students to to grph derive histogrm the qudrtic formul on their own Exmple A s well s nother exmple or two Interpret like it before histogrms nd fter stem-nd-lef they see Exmple plots A. Hve looking t the qudrtic Decide the formul s derivtion. lef plot Emphsize for given dt into setthe reltionship between ppropriteness students of histogrm explin their nd insights stem-nd- the links mong the derivtion, the constnt vlues of, b, nd the formul, nd solutions. c nd the solutions for x. ONGOING ASSESSMENT Check students fmilirity with the generl form of qudrtic nd with the projectile motion function. Also see how well they remember the qudrtic formul from previous courses. LESSON 7.4 The Qudrtic Formul 403

Discussing the Lesson LESSON EXAMPLE A This exmple revisits the sitution of Lesson 7.3, Exm ple C, but it poses problem tht reviews completing the squre nd extends it to solving n eqution. Resist the tempttion to skip this exmple nd go right to the qudrtic formul; the exmple provides specil cse for the derivtion of tht formul. In pplying the qudrtic formul to Exmple A, you might wnt to simplify to the exct 57 solutions, x 3.75. If students won t be doing the project t the end of this lesson, you might let them use the clcultor progrm found in Clcultor Note 7D. [Alert] Most mistkes mde while using clcultor with the qudrtic formul cn be blmed on not enclosing the rdicnd within prentheses, not enclosing the entire numertor within prentheses, or not enclosing the denomintor within prentheses. Guiding the Investigtion This is deepening skills investigtion. The investigtion prompts students to exmine the vlidity of their solutions in context. Students should develop this hbit of figuring out not just wht but lso if nd why. MODIFYING THE INVESTIGATION Whole Clss Complete Steps 1 through 6 with student input. Shortened Skip Step 6. One Step Remind students of the second prt of the One Step investigtion of Lesson 7.3, in which they knew the initil velocity of bsebll. Ask them when the bll hits the ground. Students cn use the vertex form they found, set it equl to 0, nd solve for time. Or they cn double the x-coordinte of the vertex. Chllenge them to solve the generl eqution x bx c 0 by completing the squre nd to come up with formul for the solution to ny qudrtic eqution. Don t be stisfied if they lredy cn reel off the qudrtic formul; the gol is for them to derive it. FACILITATING STUDENT WORK Step 1 Students re to use the projectile motion function from Lesson 7.3. If you follow the sme steps with generl qudrtic eqution, then you cn develop the qudrtic formul. This formul provides solutions to x bx c 0 in terms of, b, nd c. x bx c 0 x bx c Originl eqution. Subtrct c from both sides. x b x? c Fctor to get the leding coefficient 1. x b x b b c x b b 4 c x b b 4 4c 4 x b b 4c 4 Complete the squre. Fctor the perfect-squre trinomil on the left side nd multiply on the right side. Rewrite the right side with common denomintor. Add terms with common denomintor. x b b 4c 4 Divide both sides by. x b b 4c 4 x b b 4c x b b 4c x b _ b 4c Tke the squre root of both sides. Use the power of quotient property to tke the squre roots of the numertor nd denomintor. Subtrct b from both sides. Add terms with common denomintor. The Qudrtic Formul Given qudrtic eqution written in the form x bx c 0, the solutions re x b _ b 4c To use the qudrtic formul on the eqution in Exmple A, 16x 10x 3 0, first identify the coefficients s 16, b 10, nd c 3. The solutions re x 10 10 4(16)(3) (16) 10 1459 x _ 10 or x _ 1459 3 3 x 0.05 or x 7.55 The qudrtic formul gives you wy to find the roots of ny eqution in the form x bx c 0. The investigtion will give you n opportunity to pply the qudrtic formul in different situtions. Step 4 A grph of the function cn help students nswer this question. Step 5 In Step 3, the symmetry of the two solutions bout the x-coordinte of the vertex is consistent with the symmetry of the prbol. 404 CHAPTER 7 Qudrtic nd Other Polynomil Functions

Step 1 Step Step 3 Step 4 14; the bll reches the mximum height only once. The bll reches other Step 4 heights once on the wy up nd once on the wy down, but the top of the Step 5 bll s height, the mximum point, cn be reched only once. Step 6 EXAMPLE B Solution Investigtion How High Cn You Go? Slvdor hits bsebll t height of 3 ft nd with n initil upwrd velocity of 88 feet per second. Step 6 This step foreshdows the introduction of complex numbers in Lesson 7.5. [Alert] Wtch for students who fudge their numbers to get rel number solutions. ASSESSING PROGRESS Give specil ttention to how students re substituting, b, nd c into the qudrtic formul nd simplifying. Let x represent time in seconds fter the bll is hit, nd let y represent the height of the bll in feet. Write n eqution tht gives the height s function of time. y 16x 88x 3 Write n eqution to find the times when the bll is 4 ft bove the ground. 4 16x 88x 3 Rewrite your eqution from Step in the form x bx c 0, then use the qudrtic formul to solve. Wht is the rel-world mening of ech of your solutions? Why re there two solutions? 16x 88x 1 0; x 0.5 or x 5.5. The bll rises to 4 ft on the wy up t 0.5 s, nd it flls to 4 ft on the wy down t 5.5 s. Find the y-coordinte of the vertex of this prbol. How mny different x-vlues correspond to this y-vlue? Explin. Write n eqution to find the time when the bll reches its mximum height. Use the qudrtic formul to solve the eqution. At wht point in the solution process does it become obvious tht there is only one solution to this eqution? 16x 88x 3 14; x.75; when b 4c 0 Write n eqution to find the time when the bll reches height of 00 ft. Wht hppens when you try to solve this impossible sitution with the qudrtic formul? 16x 88x 3 00. You get the squre root of negtive number, so there re no rel solutions. It s importnt to note tht qudrtic eqution must be in the generl form x bx c 0 before you use the qudrtic formul. Solve 3x 5x 8. To use the qudrtic formul, first write the eqution in the form x bx c 0 nd identify the coefficients. 3x 5x 8 0 3, b 5, c 8 Substitute, b, nd c into the qudrtic formul. x b b 4c (5) (5) 4(3)(8) (3) 5 11 6 _ 5 11 6 x _ 5 11 6 3 8 or x _ 5 11 1 6 DISCUSSING THE INVESTIGATION [Lnguge] During student presenttions, discourge the use of the term qudrtic eqution to men qudrtic formul. The qudrtic formul cn be used to solve ll qudrtic equtions. [Criticl Question] Wht use is the qudrtic formul? [Big Ide] The qudrtic formul provides shortcut for finding the zeros of ny qudrtic function written in generl form, without hving to complete the squre. [Ask] Why re the two solutions in the form of certin mount dded to nd subtrcted from nother number? [The other number is b, the x-coordinte of the prbol s vertex. The symmetry of the prbol ensures tht its x-intercepts will be equidistnt on either side of this coordinte.] Ask whether the vlue of k, the y-vlue of the vertex, lso ppers in the qudrtic formul. The number being dded to nd subtrcted from b is the squre root of k_. Remind students tht they cn gin insight from looking t extreme cses. [Ask] When the vlue of, b, or c is 0, wht cn be sid bout the eqution, nd wht hppens to the formul? [When 0, the eqution is liner, not qudrtic, becuse the qudrtic formul requires dividing by, the result when 0 is not defined; the formul doesn t pply to liner equtions. When b 0, the eqution s grph is symmetric bout the y-xis; the formul gives two solutions equidistnt from 0 or double solution t x 0. When c 0, the eqution fctors to show root t x 0, nd the formul returns 0 s one of the roots.] [Ask] Wht if the number represented by b 4c in the squre root is 0? Both solutions re the sme, b. The prbolic grph of the qudrtic function touches the x-xis t just one point, point of tngency t the vertex: b, 0. You might describe the quntity b 4c s the discriminnt. [Ask] Wht if the number represented by b 4c is negtive? [The eqution hs no rel solutions, indicting tht its prbolic grph doesn t cross the x-xis.] You might use the Sketchpd demonstrtion Qudrtic Functions to investigte how vrious vlues of, b, nd c ffect the grph of qudrtic function. LESSON 7.4 The Qudrtic Formul 405

Discussing the Lesson LESSON EXAMPLE B You might sk students to do the clcultions on grphing clcultors to give them n opportunity to prctice inserting prentheses round the numertor nd round the terms under the rdicl. On the clcultor screen here, the verticl br is red s such tht by the clcultor, so... x 1 will replce ll x s with the vlue 1. On the TI-83/84 Plus, you would store 8_ 3 s x, use colon, nd then type in the originl eqution: 8/3 x : 3x 5x 8. As the book sys, the qudrtic formul will solve ny qudrtic eqution, but some solutions will not be rel numbers. Students will see nonrel solutions in Lesson 7.5. SUPPORT EXAMPLES 1. Use the qudrtic formul to find the zeros of the function x 5x 1. [x 0.09, 4.791]. Write the qudrtic eqution tht ws used in the qudrtic formul x 4 164()(10) y x 4x 10 Closing the Lesson 4. Restte the qudrtic formul nd perhps remind students of how it s derived from completing the squre, with or without using the formuls for h nd k. [Closing Question] Wht hppens when you pply the qudrtic formul to determine where the function y x equls 17? ASSIGNING EXERCISES Suggested Assignments: Stndrd 1 6, 8, 13 Enriched 4 9, 11, 17 Types of Exercises: Bsic 1 5 Essentil 1, 4, 5, 6, 8 Portfolio 10 Group 8 EXERCISES Review 13 17 EXERCISE NOTES Prctice Your Skills Students prctice pplying the qudrtic formul. Some students will be so intrigued by the ese of using the qudrtic formul tht they will wnt to use it with ny eqution, regrd less of its initil form. Once they re comfortble with this method, The solutions re x 8_ 3 or x 1. To check your work, substitute these vlues into the originl eqution. Here s wy to use your clcultor to check. Remember, you cn find exct solutions to some qudrtic equtions by fctoring. However, most qudrtic equtions don t fctor esily. The qudrtic formul cn be used to solve ny qudrtic eqution. 1. Rewrite ech eqution in generl form, x bx c 0. Identify, b, nd c.. 3x 13x 10 b. x 13 5x c. 3x 5x 1 d. 3x 3x 0 e. 14(x 4) (x ) (x )(x 4) x 15x 50 0; 3x 3x 0; 3, b 3, c 1, b 15, c 50. Evlute ech expression using your clcultor. Round your nswers to the nerest thousndth.. 30 30 4(5)(3) b. 30 30 4(5)(3) 0.10 5.898 (5) (5) c. 8 (8) 4(1)() d. 8 (8) 4(1)() 0.43 8.43 (1) (1) 3. Solve by ny method.. x 6x 5 0 b. x 7x 18 0 c. 5x 1x 7 0 x 1 or x 5 x or x 9 x 1 or x 1.4 4. Use the roots of the equtions in Exercise 3 to write ech of these functions in fctored form, y x r 1 x r.. y x 6x 5 b. y x 7x 18 c. y 5x 1x 7 y (x 1)(x 5) y (x )(x 9) y 5(x 1)(x 1.4) 5. Use the qudrtic formul to find the zeros of ech function.. f (x) x 7x 4 b. f (x) x 6x 3 c. y 6 x d. 5x 4 x 5. x 0.5 y x 3 6 or x 3 6 x 3 or x 4 no rel solutions Reson nd Apply You will need 6. Beth uses the qudrtic formul to solve n eqution nd gets x 9 9 4(1)(10) (1). Write the qudrtic eqution Beth strted with. x 9x 10 0 b. Write the simplified forms of the exct nswers. x 9 41 c. Wht re the x-intercepts of the grph of this qudrtic function? 9 41 A grphing clcultor for Exercises 8, 11, nd 17. nd 9 41 encourge them to lern nother method, to improve their flexibility. Exercises 1 3 [Extr Support] Hve students compre nswers on these exercises nd become mindful of common mistkes. 1. 3x 13x 10 0; 3, b 13, c 10 1b. x 5x 13 0; 1, b 5, c 13 1c. 3x 5x 1 0; 3, b 5, c 1 406 CHAPTER 7 Qudrtic nd Other Polynomil Functions

7. 5.5 7. Write qudrtic function whose grph hs these x-intercepts.. 3 nd 3 b. 4 nd 5 c. r 1 nd r y (x 3)(x 3) for 0 y (x r 1 )(x r ) for 0 8. Use the qudrtic formul to find the zeros of y x x 5. Explin wht hppens. Grph y x x 5 to confirm your observtion. How cn you recognize this sitution before using the qudrtic formul? 9. Write qudrtic function tht hs no x-intercepts. The function cn be ny qudrtic function for which b 4c is negtive. Smple nswer: y x x 1. 10. Show tht the men of the two solutions provided by the qudrtic formul is b. Explin wht this tells you bout grph. 11. These dt give the mount of wter in drining bthtub nd the mount of time fter the plug ws pulled.. Write function tht gives the mount of wter s function of time. y 4x 6.8x 49. b. How much wter ws in the tub when the plug ws pulled? c. How long did it tke the tub to empty?.76 min 1. A golden rectngle is rectngle tht cn be divided into squre nd nother smller rectngle tht is similr to the originl rectngle. In the figure t right, ABCD is golden rectngle becuse it cn be divided into squre ABFE nd rectngle FCDE, nd FCDE is similr to ABCD. Setting up proportion of the side lengths of the similr rectngles leds to. Let b 1 nd History solve this eqution for. Review 1 1 b b, 1 5 Mny people nd cultures throughout history hve felt tht the golden rectngle is one of the most visully plesing geometric shpes. It ws used in the rchitecturl designs of the Cthedrl of Notre Dme in Pris, s well s in music nd fmous works of rt. It is believed tht the erly Egyptins knew the vlue of the golden rtio (the rtio of the length of the golden rectngle 1 5 to the width) to be nd tht they used the rtio when building their pyrmids, temples, nd tombs. 13. x 14x 49 (x 7) or 13d. x 8x 8 x (14x) 49 [x (7)] (x 4x 4) 13. Complete ech eqution. 13b. x 10x 5 (x 5) (x ) or x (8x). x? 49 (x? ) b. x 10x?? 8 [x ) (4x) 4] [x ()] c. x 3x? (? ) d. x? 8 (x?? )? (x? ) x 3x 4 9 x 3 14. Find the inverse of ech function. (The inverse does not need to be function.). y (x 1) b. y (x 1) 4 c. y x x 5 y x 1 y x 4 1 y x 6 1 Exercise 4c This exercise extends the definition of fctored form to include non-integer roots. Exercise 5 As needed, remind students tht ech of these problems will hve two solutions or one double solution. Exercise 6 [Lnguge] Simplified form mens with the rithmetic performed nd the rdicl reduced, if possible. (In this exercise the rdicl cnnot be reduced.) Use prt to ssess how well students understnd the vribles in the qudrtic formul. Time (min) x Amount of wter (L) y 49. L B A [Ask] Without grphing, cn you tell whether the prbol s vertex is mximum or minimum? [It s minimum becuse is positive.] Exercise 7b You might wnt to sk students to write this eqution without frction. 7b. y (x 4) x 5 or y (x 4)(5x ) for 0 1 1.5.5 38.4 30.0 19.6 7. The Fiboncci Fountin in Bowie, Mrylnd, ws designed by mthemticin Helmn Ferguson using Fiboncci numbers nd the golden rtio. It hs 14 wter spouts rrnged horizontlly t intervls proportionl to Fiboncci numbers. Exercises 8, 9 In discovering wht hppens when they pply the qudrtic formul to n eqution with F E b C D no solution, students deepen their understnding of the reltionships mong the terms zeros, x-intercepts, nd solutions. They lso get glimpse of the ides bout complex numbers in the next lesson. 8. The solution includes the squre root of 36, so there re no rel solutions. The grph shows no x-intercepts. Before using the qudrtic function, evlute b 4c. If b 4c 0, then there will be no rel solutions. Exercise 9 This is nother opportunity to discuss the discriminnt. It cn build on conclusions reched in Exercise 8. 10. 1 b b 4c b b 4c 1 b. b The x-coordinte of the vertex, b, is midwy between the two x-intercepts. Exercise 11 This exercise requires using finite differences nd the qudrtic formul. [Alert] The plug ws pulled t time 0, not time 1. Exercise 1 The eqution cn be solved by the qudrtic formul. Only the positive solution, 1 5, mkes sense s length. [Context] History Connection Students might be interested to know how the golden rtio nd Fiboncci sequence re relted. In the Fiboncci sequence (1, 1,, 3, 5, 8,..., 4181, 6765, 10946,...) ech term is the sum of the two preceding terms. The rtio of consecutive numbers in the sequence pproches the golden rtio. 10946 1.618033999 6765 1 5 1.618033989 Rtios of lrger Fiboncci numbers will be even closer to the golden rtio. LESSON 7.4 The Qudrtic Formul 407

Exercise 16 Students my need to sketch digrm to figure out the eqution using the Pythgoren Theorem. Assume tht the wll is verticl nd the ground is horizontl. Exercise 17 Although there re 11 support cbles lbeled, their distnce prt is 1 1 of 160 ft, or 13 1_ 3 ft. Students my correctly reson tht bridge would hve two prbolic cbles, one on ech side of the rod. Their nswer of 458.3 ft is correct for the ssumption they re mking. 17. k 5.08 _ 3 ft; b j 33. _ 3 ft; c i 18.75 ft; d h 8. _ 3 ft; e g.08 _ 3 ft; f 0; totl length 9.1 _ 6 ft [Context] Engineering Connection Students who completed the Chpter 5 explortion The Number e might be interested to know tht the eqution for ctenry curve contins the number e. y e x/ e x/ 7. 15. Convert these qudrtic functions to generl form.. y (x 3)(x 5) y x x 15 b. y (x 1) 4 16. A 0 ft ldder lens ginst building. Let x represent the distnce between the building nd the foot of the ldder, nd let y represent the height the ldder reches on the building.. Write n eqution for y in terms of x. b. Find the height the ldder reches on the building if the foot of the ldder is 10 ft from the building. pproximtely 17.3 ft c. Find the distnce of the foot of the ldder from the building if the ldder must rech 18 ft up the wll. pproximtely 8.7 ft 17. APPLICATION The min cbles of suspension bridge typiclly hng in the shpe of prllel prbols on both sides of the rodwy. The verticl support cbles, lbeled k, re eqully spced, nd the center of the prbolic cble touches the rodwy t f. If this bridge hs spn of 160 ft between towers, nd the towers rech height of 75 ft bove the rod, wht is the length of ech support cble, k? Wht is the totl length of verticl support cble needed for the portion of the bridge between the two towers? Engineering y 400 x The rodwy of suspension bridge is suspended, or hngs, from lrge steel support cbles. By itself, cble hngs in the shpe of ctenry curve. However, with the weight of rodwy ttched, the curvture chnges, nd the cble hngs in prbolic curve. It is importnt for engineers to ensure tht cbles re the correct lengths to mke level rodwy. b c d e y x 4x y x 0 ft A chin hngs in the shpe of ctenry curve. f g h i j k The grph of this function is symmetric bout the y-xis; is the y-intercept. EXTENSION Hve students use geometry softwre to nlyze wht hppens to the vertex of prbol when you vry in the eqution of the generl form but hold b nd c constnt. Students then cn describe wht hppens when b vries nd nd c re held constnt. This could lso be done with Fthom or the TI-Nspire. CALCULATOR PROGRAM FOR THE QUADRATIC FORMULA Write clcultor progrm tht uses the qudrtic formul to solve equtions. The progrm should clculte nd disply the qud(1,-1,-6) two solutions for qudrtic eqution in the form x bx c. Depending on the type of clcultor you hve, the user cn give the, b, nd c vlues s prmeters for the progrm (s shown) or the progrm cn prompt the user for those vlues. Your project should include A written record of the steps your progrm uses. An explntion of how the progrm works. The results of solving t lest two equtions by hnd nd with your progrm to verify tht your progrm works. 3 - Done Supporting the If you don t wnt to chllenge students to devise their own progrms, you cn give them Clcultor Note 7D, which contins smple progrm. OUTCOMES The report includes simple progrm tht gives the solutions s deciml pproximtions. Documenttion is provided to confirm tht the progrm works. The student hs creted progrm tht cn give solutions s exct vlues in the form of reduced rdicl expressions. The student hs creted progrm tht checks the discriminnt to tell the user when the number of roots is 0, 1, or. 408 CHAPTER 7 Qudrtic nd Other Polynomil Functions