SOLVING EQUATIONS BY FACTORING


 Judith Black
 1 years ago
 Views:
Transcription
1 316 (560) Chpter 5 Exponents nd Polynomils 5.9 SOLVING EQUATIONS BY FACTORING In this setion The Zero Ftor Property Applitions helpful hint Note tht the zero ftor property is our seond exmple of getting n equivlent eqution without doing the sme thing to eh side. Wht ws the first? E X A M P L E 1 The tehniques of ftoring n e used to solve equtions involving polynomils tht nnot e solved y the other methods tht you hve lerned. After you lern to solve equtions y ftoring, we will use this tehnique to solve some new pplied prolems in this setion nd in Chpters 6 nd 8. The Zero Ftor Property The eqution 0 indites tht the produt of two unknown numers is 0. But the produt of two rel numers is zero only when one or the other of the numers is 0. So even though we do not know extly the vlues of nd from 0, we do know tht 0 or 0. This ide is lled the zero ftor property. Zero Ftor Property The eqution 0 is equivlent to the ompound eqution 0 or 0. The next exmple shows how to use the zero ftor property to solve n eqution in one vrile. Using the zero ftor property Solve x 2 x We ftor the lefthnd side of the eqution to get produt of two ftors tht re equl to 0. Then we write n equivlent eqution using the zero ftor property. x 2 x 12 0 (x 4)(x 3) 0 Ftor the lefthnd side. x 4 0 or x 3 0 Zero ftor property x 4 or x 3 Solve eh prt of the ompound eqution. Chek tht oth 4 nd 3 stisfy x 2 x If x 4, we get ( 4) 2 ( 4) If x 3, we get (3) So the solution set is 4, 3. The zero ftor property is used only in solving polynomil equtions tht hve zero on one side nd polynomil tht n e ftored on the other side. The polynomils tht we ftored most often were the qudrti polynomils. The equtions tht we will solve most often using the zero ftor property will e qudrti equtions. Qudrti Eqution If,, nd re rel numers, with 0, then the eqution x 2 x 0 is lled qudrti eqution.
2 5.9 Solving Equtions y Ftoring (561) 317 M A T H A T W O R K Sems Merdo, professionl odyorder nd 1988 Ntionl Chmpion, hrges the wves off Hwii, Thiti, Indonesi, Mexio, nd Cliforni. In hoosing ord for ompetition nd for the mneuvers he wnts to perform, Merdo ftors in his height nd weight s well s the size, power, nd temperture of the wves he will e riding. In older wter softer, more flexile ord is used; in wrmer wter stiffer ord is hosen. When wves rsh on shore, the ride usully lsts 3 to 5 seonds, nd shorter ord with nrrow til is hosen for greter ontrol. When wves rek long snd r or reef, the ride n sometimes lst s long s 2 minutes, nd strighter ord with more surfe re is hosen so tht the ord will move fster nd llow the rider to pull off more mneuvers. Bsi mneuvers inlude ottom turns, erils, forwrd nd reverse 360 s, nd el rollos. As one of the top 10 odyorders in the world, Merdo helps to design the ords he uses. Performne levels re gretly inresed with finetuned equipment nd tehniques, he sys. In Exerise 63 of Setion 5.9 you will find the dimensions of given odyord. BODYBOARD DESIGNER In Chpter 8 we will study qudrti equtions further nd solve qudrti equtions tht nnot e solved y ftoring. Keep the following strtegy in mind when solving equtions y ftoring. study tip We re ll retures of hit. When you find ple in whih you study suessfully, stik with it. Using the sme ple for studying will help you to onentrte nd to ssoite the ple with good studying. Strtegy for Solving Equtions y Ftoring 1. Write the eqution with 0 on the righthnd side. 2. Ftor the lefthnd side. 3. Use the zero ftor property to get simpler equtions. (Set eh ftor equl to 0.) 4. Solve the simpler equtions. 5. Chek the nswers in the originl eqution. E X A M P L E 2 Solving qudrti eqution y ftoring Solve eh eqution. ) 10x 2 5x ) 3x 6x 2 9 ) Use the steps in the strtegy for solving equtions y ftoring: 10x 2 5x Originl eqution 10x 2 5x 0 Rewrite with zero on the righthnd side. 5x(2x 1) 0 Ftor the lefthnd side. 5x 0 or 2x 1 0 Zero ftor property x 0 or x 1 Solve for x. 2 The solution set is 0, 1 2. Chek eh solution in the originl eqution.
3 318 (562) Chpter 5 Exponents nd Polynomils ) First rewrite the eqution with 0 on the righthnd side nd the lefthnd side in order of desending exponents: 3x 6x 2 9 Originl eqution 6x 2 3x 9 0 Add 9 to eh side. 2x 2 x 3 0 Divide eh side y 3. (2x 3)(x 1) 0 Ftor. 2x 3 0 or x 1 0 Zero ftor property x 3 or x 1 Solve for x. 2 The solution set is 1, 3 2. Chek eh solution in the originl eqution. CAUTION If we divide eh side of 10x 2 5x y 5x, we get 2x 1, or x 1 2. We do not get x 0. By dividing y 5x we hve lost one of the ftors nd one of the solutions. In the next exmple there re more thn two ftors, ut we n still write n equivlent eqution y setting eh ftor equl to 0. E X A M P L E 3 lultor loseup To hek, use Y= to enter y 1 2x 3 3x 2 8x 12. Then use the vriles feture (VARS) to find y 1 ( 2), y 1 (3 2), nd y 1 (2). Solving ui eqution y ftoring Solve 2x 3 3x 2 8x First notie tht the first two terms hve the ommon ftor x 2 nd the lst two terms hve the ommon ftor 4. x 2 (2x 3) 4(2x 3) 0 Ftor y grouping. (x 2 4)(2x 3) 0 Ftor out 2x 3. (x 2)(x 2)(2x 3) 0 Ftor ompletely. x 2 0 or x 2 0 or 2x 3 0 Set eh ftor equl to 0. x 2 or x 2 or x 3 2 The solution set is 2, 3 2, 2. Chek eh solution in the originl eqution. The eqution in the next exmple involves solute vlue. E X A M P L E 4 Solving n solute vlue eqution y ftoring Solve x 2 2x First write n equivlent ompound eqution without solute vlue: x 2 2x 16 8 or x 2 2x 16 8 x 2 2x 24 0 or x 2 2x 8 0 (x 6)(x 4) 0 or (x 4)(x 2) 0 x 6 0 or x 4 0 or x 4 0 or x 2 0 x 6 or x 4 or x 4 or x 2 The solution set is 2, 4, 4, 6. Chek eh solution.
4 Applitions 5.9 Solving Equtions y Ftoring (563) 319 Mny pplied prolems n e solved y using equtions suh s those we hve een solving. E X A M P L E 5 helpful hint To prove the Pythgoren theorem, drw two squres with sides of length, nd prtition them s shown. Are of room Ronld s living room is 2 feet longer thn it is wide, nd its re is 168 squre feet. Wht re the dimensions of the room? Let x e the width nd x 2 e the length. See Figure 5.1. Beuse the re of retngle is the length times the width, we n write the eqution x(x 2) Erse the four tringles in eh piture. Sine we strted with equl res, we must hve equl res fter ersing the tringles: x FIGURE 5.1 x 2 We solve the eqution y ftoring: x 2 2x (x 12)(x 14) 0 x 12 0 or x 14 0 x 12 or x 14 Beuse the width of room is positive numer, we disregrd the solution x 14. We use x 12 nd get width of 12 feet nd length of 14 feet. Chek this nswer y multiplying 12 nd 14 to get 168. Applitions involving qudrti equtions often require theorem lled the Pythgoren theorem. This theorem sttes tht in ny right tringle the sum of the squres of the lengths of the legs is equl to the length of the hypotenuse squred. The Pythgoren Theorem The tringle shown is right tringle if nd only if Hypotenuse Legs We use the Pythgoren theorem in the next exmple.
5 320 (564) Chpter 5 Exponents nd Polynomils E X A M P L E 6 5 x 7 x FIGURE 5.2 Using the Pythgoren theorem Shirley used 14 meters of fening to enlose retngulr region. To e sure tht the region ws retngle, she mesured the digonls nd found tht they were 5 meters eh. (If the opposite sides of qudrilterl re equl nd the digonls re equl, then the qudrilterl is retngle.) Wht re the length nd width of the retngle? The perimeter of retngle is twie the length plus twie the width, P 2L 2W. Beuse the perimeter is 14 meters, the sum of one length nd one width is 7 meters. If we let x represent the width, then 7 x is the length. We use the Pythgoren theorem to get reltionship mong the length, width, nd digonl. See Figure 5.2. x 2 (7 x) x x x x 2 14x 24 0 Pythgoren theorem Simplify. Simplify. x 2 7x 12 0 Divide eh side y 2. (x 3)(x 4) 0 Ftor the lefthnd side. x 3 0 or x 4 0 Zero ftor property x 3 or x 4 7 x 4 or 7 x 3 Solving the eqution gives two possile retngles: 3 y 4 retngle or 4 y 3 retngle. However, those re identil retngles. The retngle is 3 meters y 4 meters. WARMUPS True or flse? Explin your nswer. 1. The eqution (x 1)(x 3) 12 is equivlent to x 1 3 or x 3 4. Flse 2. Equtions solved y ftoring my hve two solutions. True 3. The eqution d 0 is equivlent to 0 or d 0. True 4. The eqution x is equivlent to the ompound eqution x or x Flse 5. The solution set to the eqution (2x 1)(3x 4) 0 is 1 2, 4 3. True 6. The Pythgoren theorem sttes tht the sum of the squres of ny two sides of ny tringle is equl to the squre of the third side. Flse 7. If the perimeter of retngulr room is 38 feet, then the sum of the length nd width is 19 feet. True 8. Two numers tht hve sum of 8 n e represented y x nd 8 x. True 9. The solution set to the eqution x(x 1)(x 2) 0is 1, 2. Flse 10. The solution set to the eqution 3(x 2)(x 5) 0 is 3, 2, 5. Flse
6 5.9 Solving Equtions y Ftoring (565) EXERCISES Reding nd Writing After reding this setion, write out the nswers to these questions. Use omplete sentenes. 1. Wht is the zero ftor property? The zero ftor property sys tht if 0 then either 0 or Wht is qudrti eqution? A qudrti eqution is n eqution of the form x 2 x 0 with Where is the hypotenuse in right tringle? The hypotenuse of right tringle is the side opposite the right ngle. 4. Where re the legs in right tringle? The legs of right tringle re the sides tht form the right ngle. 5. Wht is the Pythgoren theorem? The Pythgoren theorem sys tht tringle is right tringle if nd only if the sum of the squres of the legs is equl to the squre of the hypotenuse. 6. Where is the digonl of retngle? The digonl of retngle is the line segment tht joins two opposite verties. Solve eh eqution. See Exmples (x 5)(x 4) 0 4, 5 8. ( 6)( 5) 0 5, 6 9. (2x 5)(3x 4) 0 5 2, (3k 8)(4k 3) 0 8 3, w 2 5w , t 2 6t , m 2 7m 0 0, h 2 5h 0 0, , p 2 p 42 6, x 2 3x , x 2 16x , z z 10 4, m m 2 3 3, x 3 4x 0 2, 0, x x 3 0 4, 0, w 3 4w 2 25w , 4, , 2, n 3 2n 2 n 2 0 1, 1, w 3 w 2 25w , 1, 5 Solve eh eqution. See Exmple x , 1, 1, x , 3, 3, x 2 2x , 6, 4, x 2 2x , 3, 1, x 2 4x 2 2 4, 2, x 2 8x 8 8 8, 4, x 2 6x 1 8 7, 3, x 2 x , 3, 4, 6 Solve eh eqution x 2 x 6 3 2, x 2 14x 5 5, x 2 5x 6 6, 3, 2, x 2 6x , 4, 2, x 2 5x 6 6, x 5x (x 2)(x 1) 12 5, (x 2)(x 3) 20 7, y 3 9y 2 20y 0 5, 4, m 3 2m 2 3m 0 1, 0, , 0, x 3 125x 5, 0, (2x 1)(x 2 9) 0 3, 1 2,3 48. (x 1)(x 3)(x 9) 0 3, 1, x 2 12x x 2 8x Solve eh eqution for y. Assume nd re positive numers. 51. y 2 y 0 0, 52. y 2 y y 0, y 2 2 0, 54. 9y 2 6y y 2 4y y 2 2 0, 57. y 2 3y y 3 3, y 2 2y 2 0 Solve eh prolem. See Exmples 5 nd Color print. The length of new super size olor print is 2 inhes more thn the width. If the re is 24 squre inhes, wht re the length nd width? Width 4 inhes, length 6 inhes
7 322 (566) Chpter 5 Exponents nd Polynomils 60. Tennis ourt dimensions. In singles ompetition, eh plyer plys on retngulr re of 117 squre yrds. Given tht the length of tht re is 4 yrds greter thn its width, find the length nd width. Width 9 yrds, length 13 yrds 61. Missing numers. The sum of two numers is 13 nd their produt is 36. Find the numers. 4 nd More missing numers. The sum of two numers is 6.5, nd their produt is 9. Find the numers. 2 nd Bodyording. The Sems Chnnel pro odyord shown in the figure hs length tht is 21 inhes greter thn its width. Any rider weighing up to 200 pounds n use it euse its surfe re is 946 squre inhes. Find the length nd width. Length 43 inhes, width 22 inhes x 21 in. Height (feet) ) Use the ompnying grph to estimte the mximum height rehed y the rrow. 64 feet d) At wht time does the rrow reh its mximum height? 2 seonds Time (seonds) FIGURE FOR EXERCISE Time until impt. If n ojet is dropped from height of s 0 feet, then its ltitude fter t seonds is given y the formul S 16t 2 s 0. If pk of emergeny supplies is dropped from n irplne t height of 1600 feet, then how long does it tke for it to reh the ground? 10 seonds 67. Yolnd s loset. The length of Yolnd s loset is 2 feet longer thn twie its width. If the digonl mesures 13 feet, then wht re the length nd width? Width 5 feet, length 12 feet x in. FIGURE FOR EXERCISE New dimensions in grdening. Mry Gold hs retngulr flower ed tht mesures 4 feet y 6 feet. If she wnts to inrese the length nd width y the sme mount to hve flower ed of 48 squre feet, then wht will e the new dimensions? 6 feet y 8 feet 2x 2 ft 13 ft x ft 4 ft 6 ft x ft FIGURE FOR EXERCISE Ski jump. The se of ski rmp forms right tringle. One leg of the tringle is 2 meters longer thn the other. If the hypotenuse is 10 meters, then wht re the lengths of the legs? 6 feet nd 8 feet x ft FIGURE FOR EXERCISE Shooting rrows. An rher shoots n rrow stright upwrd t 64 feet per seond. The height of the rrow h(t) (in feet) t time t seonds is given y the funtion h(t) 16t 2 64t. ) Use the ompnying grph to estimte the mount of time tht the rrow is in the ir. 4 seonds ) Algerilly find the mount of time tht the rrow is in the ir. 4 seonds 10 m x + 2 m x m FIGURE FOR EXERCISE Trimming gte. A totl of 34 feet of 1 4 lumer is used round the perimeter of the gte shown in the figure on the next pge. If the digonl re is 13 feet long, then wht re the length nd width of the gte? Width 5 feet, length 12 feet
8 Chpter 5 Collortive Ativities (567) ft 74. Arrnging the rows. Mr. Converse hs 112 students in his lger lss with n equl numer in eh row. If he rrnges the desks so tht he hs one fewer rows, he will hve two more students in eh row. How mny rows did he hve originlly? 8 GETTING MORE INVOLVED FIGURE FOR EXERCISE Perimeter of retngle. The perimeter of retngle is 28 inhes, nd the digonl mesures 10 inhes. Wht re the length nd width of the retngle? Length 8 inhes, width 6 inhes 71. Conseutive integers. The sum of the squres of two onseutive integers is 25. Find the integers. 3 nd 4, or 4 nd Pete s grden. Eh row in Pete s grden is 3 feet wide. If the rows run north nd south, he n hve two more rows thn if they run est nd west. If the re of Pete s grden is 135 squre feet, then wht re the length nd width? Length 15 feet, width 9 feet 73. House plns. In the plns for their drem house the Bileys hve mster edroom tht is 240 squre feet in re. If they inrese the width y 3 feet, they must derese the length y 4 feet to keep the originl re. Wht re the originl dimensions of the edroom? Length 20 feet, width 12 feet 75. Writing. If you divide eh side of x 2 x y x, you get x 1. If you sutrt x from eh side of x 2 x, you get x 2 x 0, whih hs two solutions. Whih method is orret? Explin. 76. Coopertive lerning. Work with group to exmine the following solution to x 2 2x 1: x(x 2) 1 x 1 or x 2 1 x 1 or x 1 Is this method orret? Explin. 77. Coopertive lerning. Work with group to exmine the following steps in the solution to 5x (x 2 1) 0 5(x 1)(x 1) 0 x 1 0 or x 1 0 x 1 or x 1 Wht hppened to the 5? Explin. COLLABORATIVE ACTIVITIES Mgi Triks Jim nd Sdr re tlking one dy fter lss. Sdr: Jim, I hve trik for you. Think of numer etween 1 nd 10. I will sk you to do some things to this numer. Then t the end tell me your result, nd I will tell you your numer. Jim: Oh, yeh you proly rig it so the result is my numer. Sdr: Come on Jim, give it try nd see. Jim: Oky, oky, I thought of numer. Sdr: Good, now write it down, nd don t let me see your pper. Now dd x. Got tht? Now multiply everything y 2. Jim: Hey, I didn t know you were going to mke me think! This is lger! Sdr: I know, now just do it. Oky, now squre the polynomil. Got tht? Now sutrt 4x 2. Jim: How did you know I hd 4x 2? I told you this ws rigged! Sdr: Of ourse it s rigged, or it wouldn t work. Do you wnt to finish or not? Jim: Yeh, I guess so. Go hed, wht do I do next? Sdr: Divide y 4. Oky, now sutrt the xterm. Jim: Just ny old xterm? Got ny prtiulr oeffiient in mind? Grouping: Two students per group Topi: Prtie with exponent rules, multiplying polynomils Sdr: Now stop tesing me. I know you only hve one xterm left, so sutrt it. Jim: H, h, I ould give you hint out the oeffiient, ut tht wouldn t e fir, would it? Sdr: Well you ould, nd then I ould tell you your numer, or you ould just tell me the numer you hve left fter sutrting. Jim: Oky, the numer I hd left t the end ws 25. Let s see if you n tell me wht the oeffiient of the xterm I sutrted is. Sdr: Ah, then the numer you hose t the eginning ws 5, nd the oeffiient ws 10! Jim: Hey, you re right! How did you do tht? In your group, follow Sdr s instrutions nd determine why she knew Jim s numer. Mke up nother set of instrutions to use s mgi trik. Be sure to use vriles nd some of the exponent rules or rules for multiplying polynomils tht you lerned in this hpter. Exhnge instrutions with nother group nd see whether you n figure out how their trik works.
SOLVING QUADRATIC EQUATIONS BY FACTORING
6.6 Solving Qudrti Equtions y Ftoring (6 31) 307 In this setion The Zero Ftor Property Applitions 6.6 SOLVING QUADRATIC EQUATIONS BY FACTORING The tehniques of ftoring n e used to solve equtions involving
More informationThe area of the larger square is: IF it s a right triangle, THEN + =
8.1 Pythgoren Theorem nd 2D Applitions The Pythgoren Theorem sttes tht IF tringle is right tringle, THEN the sum of the squres of the lengths of the legs equls the squre of the hypotenuse lengths. Tht
More informationThe remaining two sides of the right triangle are called the legs of the right triangle.
10 MODULE 6. RADICAL EXPRESSIONS 6 Pythgoren Theorem The Pythgoren Theorem An ngle tht mesures 90 degrees is lled right ngle. If one of the ngles of tringle is right ngle, then the tringle is lled right
More informationThree squares with sides 3, 4, and 5 units are used to form the right triangle shown. In a right triangle, the sides have special names.
1 The Pythgoren Theorem MAIN IDEA Find length using the Pythgoren Theorem. New Voulry leg hypotenuse Pythgoren Theorem Mth Online glenoe.om Extr Exmples Personl Tutor SelfChek Quiz Three squres with
More information4.5 The Converse of the
Pge 1 of. The onverse of the Pythgoren Theorem Gol Use the onverse of Pythgoren Theorem. Use side lengths to lssify tringles. Key Words onverse p. 13 grdener n use the onverse of the Pythgoren Theorem
More informationMATH PLACEMENT REVIEW GUIDE
MATH PLACEMENT REVIEW GUIDE This guie is intene s fous for your review efore tking the plement test. The questions presente here my not e on the plement test. Although si skills lultor is provie for your
More informationD e c i m a l s DECIMALS.
D e i m l s DECIMALS www.mthletis.om.u Deimls DECIMALS A deiml numer is sed on ple vlue. 214.84 hs 2 hundreds, 1 ten, 4 units, 8 tenths nd 4 hundredths. Sometimes different 'levels' of ple vlue re needed
More information1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5.
. Definition, Bsi onepts, Types. Addition nd Sutrtion of Mtries. Slr Multiplition. Assignment nd nswer key. Mtrix Multiplition. Assignment nd nswer key. Determinnt x x (digonl, minors, properties) summry
More information11. PYTHAGORAS THEOREM
11. PYTHAGORAS THEOREM 111 Along the Nile 2 112 Proofs of Pythgors theorem 3 113 Finding sides nd ngles 5 114 Semiirles 7 115 Surds 8 116 Chlking hndll ourt 9 117 Pythgors prolems 10 118 Designing
More informationEssential Question What are the Law of Sines and the Law of Cosines?
9.7 TEXS ESSENTIL KNOWLEDGE ND SKILLS G.6.D Lw of Sines nd Lw of osines Essentil Question Wht re the Lw of Sines nd the Lw of osines? Disovering the Lw of Sines Work with prtner.. opy nd omplete the tle
More informationProving the Pythagorean Theorem
Proving the Pythgoren Theorem Proposition 47 of Book I of Eulid s Elements is the most fmous of ll Eulid s propositions. Disovered long efore Eulid, the Pythgoren Theorem is known y every high shool geometry
More information81. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary
81 The Pythgoren Theorem nd Its Converse Voulry Review 1. Write the squre nd the positive squre root of eh numer. Numer Squre Positive Squre Root 9 81 3 1 4 1 16 1 2 Voulry Builder leg (noun) leg Relted
More informationFractions: Arithmetic Review
Frtions: Arithmeti Review Frtions n e interprete s rtios omprisons of two quntities. For given numer expresse in frtion nottion suh s we ll the numertor n the enomintor n it is helpful to interpret this
More informationRight Triangle Trigonometry 8.7
304470_Bello_h08_se7_we 11/8/06 7:08 PM Pge R1 8.7 Right Tringle Trigonometry R1 8.7 Right Tringle Trigonometry T E G T I N G S T R T E D The origins of trigonometry, from the Greek trigonon (ngle) nd
More informationThank you for participating in Teach It First!
Thnk you for prtiipting in Teh It First! This Teh It First Kit ontins Common Core Coh, Mthemtis teher lesson followed y the orresponding student lesson. We re onfident tht using this lesson will help you
More informationLesson 18.2: Right Triangle Trigonometry
Lesson 8.: Right Tringle Trigonometry lthough Trigonometry is used to solve mny prolems, historilly it ws first pplied to prolems tht involve right tringle. This n e extended to nonright tringles (hpter
More informationexcenters and excircles
21 onurrene IIi 2 lesson 21 exenters nd exirles In the first lesson on onurrene, we sw tht the isetors of the interior ngles of tringle onur t the inenter. If you did the exerise in the lst lesson deling
More informationSECTION 72 Law of Cosines
516 7 Additionl Topis in Trigonometry h d sin s () tn h h d 50. Surveying. The lyout in the figure t right is used to determine n inessile height h when seline d in plne perpendiulr to h n e estlished
More information11.1 Conic sections (conics)
. Coni setions onis Coni setions re formed the intersetion of plne with right irulr one. The tpe of the urve depends on the ngle t whih the plne intersets the surfe A irle ws studied in lger in se.. We
More information10.3 Systems of Linear Equations: Determinants
758 CHAPTER 10 Systems of Equtions nd Inequlities 10.3 Systems of Liner Equtions: Determinnts OBJECTIVES 1 Evlute 2 y 2 Determinnts 2 Use Crmer s Rule to Solve System of Two Equtions Contining Two Vriles
More informationState the size of angle x. Sometimes the fact that the angle sum of a triangle is 180 and other angle facts are needed. b y 127
ngles 2 CHTER 2.1 Tringles Drw tringle on pper nd lel its ngles, nd. Ter off its orners. Fit ngles, nd together. They mke stright line. This shows tht the ngles in this tringle dd up to 180 ut it is not
More informationEnd of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below.
End of term: TEST A You will need penil nd ruler. Yer Nme Clss Dte Complete the missing numers in the sequenes elow. 8 30 3 28 2 9 25 00 75 25 2 Put irle round ll of the following shpes whih hve 3 shded.
More informationSect 8.3 Triangles and Hexagons
13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed twodimensionl geometric figure consisting of t lest three line segments for its
More information15. Let f (x) = 3x Suppose rx 2 + sx + t = 0 where r 0. Then x = 24. Solve 5x 25 < 20 for x. 26. Let y = 7x
Pretest Review The pretest will onsist of 0 problems, eh of whih is similr to one of the following 49 problems If you n do problems like these 49 listed below, you will hve no problem with the pretest
More informationChapter15 SAMPLE. Simultaneous equations. Contents: A B C D. Graphical solution Solution by substitution Solution by elimination Problem solving
Chpter15 Simultneous equtions Contents: A B C D Grphil solution Solution y sustitution Solution y elimintion Prolem solving 308 SIMULTANEOUS EQUATIONS (Chpter 15) Opening prolem Ewen wnts to uy pie, ut
More informationPROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * challenge questions
PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * hllenge questions e The ll will strike the ground 1.0 s fter it is struk. Then v x = 20 m s 1 nd v y = 0 + (9.8 m s 2 )(1.0 s) = 9.8 m s 1 The speed
More informationKnow the sum of angles at a point, on a straight line and in a triangle
2.1 ngle sums Know the sum of ngles t point, on stright line n in tringle Key wors ngle egree ngle sum n ngle is mesure of turn. ngles re usully mesure in egrees, or for short. ngles tht meet t point mke
More information8.2 Trigonometric Ratios
8.2 Trigonometri Rtios Ojetives: G.SRT.6: Understnd tht y similrity, side rtios in right tringles re properties of the ngles in the tringle, leding to definitions of trigonometri rtios for ute ngles. For
More informationGeometry 71 Geometric Mean and the Pythagorean Theorem
Geometry 71 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the
More informationRatio and Proportion
Rtio nd Proportion Rtio: The onept of rtio ours frequently nd in wide vriety of wys For exmple: A newspper reports tht the rtio of Repulins to Demorts on ertin Congressionl ommittee is 3 to The student/fulty
More informationTHE PYTHAGOREAN THEOREM
THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most wellknown nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this
More informationISTM206: Lecture 3 Class Notes
IST06: Leture 3 Clss otes ikhil Bo nd John Frik 9905 Simple ethod. Outline Liner Progrmming so fr Stndrd Form Equlity Constrints Solutions, Etreme Points, nd Bses The Representtion Theorem Proof of the
More informationLesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle
: Using Trigonometry to Find Side Lengths of n Aute Tringle Clsswork Opening Exerise. Find the lengths of d nd e.. Find the lengths of x nd y. How is this different from prt ()? Exmple 1 A surveyor needs
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationChapter. Contents: A Constructing decimal numbers
Chpter 9 Deimls Contents: A Construting deiml numers B Representing deiml numers C Deiml urreny D Using numer line E Ordering deimls F Rounding deiml numers G Converting deimls to frtions H Converting
More informationWords Symbols Diagram. abcde. a + b + c + d + e
Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To
More informationChess and Mathematics
Chess nd Mthemtis in UK Seondry Shools Dr Neill Cooper Hed of Further Mthemtis t Wilson s Shool Mnger of Shool Chess for the English Chess Federtion Mths in UK Shools KS (up to 7 yers) Numers: 5 + 7; x
More informationMaximum area of polygon
Mimum re of polygon Suppose I give you n stiks. They might e of ifferent lengths, or the sme length, or some the sme s others, et. Now there re lots of polygons you n form with those stiks. Your jo is
More informationThe Pythagorean Theorem Tile Set
The Pythgoren Theorem Tile Set Guide & Ativities Creted y Drin Beigie Didx Edution 395 Min Street Rowley, MA 01969 www.didx.om DIDAX 201 #211503 1. Introdution The Pythgoren Theorem sttes tht in right
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationAngles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example
2.1 Angles Reognise lternte n orresponing ngles Key wors prllel lternte orresponing vertilly opposite Rememer, prllel lines re stright lines whih never meet or ross. The rrows show tht the lines re prllel
More informationRight Triangle Trigonometry
CONDENSED LESSON 1.1 Right Tringle Trigonometr In this lesson ou will lern out the trigonometri rtios ssoited with right tringle use trigonometri rtios to find unknown side lengths in right tringle use
More informationFunctions A B C D E F G H I J K L. Contents:
Funtions Contents: A reltion is n set of points whih onnet two vriles. A funtion, sometimes lled mpping, is reltion in whih no two different ordered pirs hve the sme oordinte or first omponent. Algeri
More informationLISTENING COMPREHENSION
PORG, přijímí zkoušky 2015 Angličtin B Reg. číslo: Inluded prts: Points (per prt) Points (totl) 1) Listening omprehension 2) Reding 3) Use of English 4) Writing 1 5) Writing 2 There re no extr nswersheets
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationSimple Electric Circuits
Simple Eletri Ciruits Gol: To uild nd oserve the opertion of simple eletri iruits nd to lern mesurement methods for eletri urrent nd voltge using mmeters nd voltmeters. L Preprtion Eletri hrges move through
More informationThe Math Learning Center PO Box 12929, Salem, Oregon 97309 0929 Math Learning Center
Resource Overview Quntile Mesure: Skill or Concept: 1010Q Determine perimeter using concrete models, nonstndrd units, nd stndrd units. (QT M 146) Use models to develop formuls for finding res of tringles,
More informationThe Pythagorean Theorem
The Pythgoren Theorem Pythgors ws Greek mthemtiin nd philosopher, orn on the islnd of Smos (. 58 BC). He founded numer of shools, one in prtiulr in town in southern Itly lled Crotone, whose memers eventully
More informationOVERVIEW Prove & Use the Laws of Sines & Cosines G.SRT.10HONORS
OVERVIEW Prove & Use te Lws of Sines & osines G.SRT.10HONORS G.SRT.10 (HONORS ONLY) Prove te Lws of Sines nd osines nd use tem to solve prolems. No interprettion needed  prove te Lw of Sines nd te Lw
More informationa 2 + b 2 = c 2. There are many proofs of this theorem. An elegant one only requires that we know that the area of a square of side L is L 2
Pythgors Pythgors A right tringle, suh s shown in the figure elow, hs one 90 ngle. The long side of length is the hypotenuse. The short leg (or thetus) hs length, nd the long leg hs length. The theorem
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationIntroduction. Law of Cosines. a 2 b2 c 2 2bc cos A. b2 a 2 c 2 2ac cos B. c 2 a 2 b2 2ab cos C. Example 1
3330_060.qxd 1/5/05 10:41 M Pge 439 Setion 6. 6. Lw of osines 439 Lw of osines Wht you should lern Use the Lw of osines to solve olique tringles (SSS or SS). Use the Lw of osines to model nd solve rellife
More informationHeron s Formula for Triangular Area
Heron s Formul for Tringulr Are y Christy Willims, Crystl Holom, nd Kyl Gifford Heron of Alexndri Physiist, mthemtiin, nd engineer Tught t the museum in Alexndri Interests were more prtil (mehnis, engineering,
More informationThe AVL Tree Rotations Tutorial
The AVL Tree Rottions Tutoril By John Hrgrove Version 1.0.1, Updted Mr222007 Astrt I wrote this doument in n effort to over wht I onsider to e drk re of the AVL Tree onept. When presented with the tsk
More informationP.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn
33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of
More informationSquare Roots Teacher Notes
Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this
More informationPractice Test 2. a. 12 kn b. 17 kn c. 13 kn d. 5.0 kn e. 49 kn
Prtie Test 2 1. A highwy urve hs rdius of 0.14 km nd is unnked. A r weighing 12 kn goes round the urve t speed of 24 m/s without slipping. Wht is the mgnitude of the horizontl fore of the rod on the r?
More informationTallahassee Community College. Simplifying Radicals
Tllhssee Communit College Simplifing Rdils The squre root of n positive numer is the numer tht n e squred to get the numer whose squre root we re seeking. For emple, 1 euse if we squre we get 1, whih is
More informationFinal Exam covers: Homework 0 9, Activities 1 20 and GSP 1 6 with an emphasis on the material covered after the midterm exam.
MTH 494.594 / FINL EXM REVIEW Finl Exm overs: Homework 0 9, tivities 1 0 nd GSP 1 6 with n emphsis on the mteril overed fter the midterm exm. You my use oth sides of one 3 5 rd of notes on the exm onepts
More informationLecture 15  Curve Fitting Techniques
Lecture 15  Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting  motivtion For root finding, we used given function to identify where it crossed zero where does fx
More informationThe Parallelogram Law. Objective: To take students through the process of discovery, making a conjecture, further exploration, and finally proof.
The Prllelogrm Lw Objective: To tke students through the process of discovery, mking conjecture, further explortion, nd finlly proof. I. Introduction: Use one of the following Geometer s Sketchpd demonstrtion
More informationLesson 2.1 Inductive Reasoning
Lesson.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 1, 16,,. 400, 00, 100, 0,,,. 1 8, 7, 1, 4,, 4.,,, 1, 1, 0,,. 60, 180, 10,
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationThe Quadratic Formula and the Discriminant
99 The Qudrtic Formul nd the Discriminnt Objectives Solve qudrtic equtions by using the Qudrtic Formul. Determine the number of solutions of qudrtic eqution by using the discriminnt. Vocbulry discriminnt
More informationFor the Final Exam, you will need to be able to:
Mth B Elementry Algebr Spring 0 Finl Em Study Guide The em is on Wednesdy, My 0 th from 7:00pm 9:0pm. You re lloed scientific clcultor nd " by 6" inde crd for notes. On your inde crd be sure to rite ny
More information2 If a branch is prime, no other factors
Chpter 2 Multiples, nd primes 59 Find the prime of 50 by drwing fctor tree. b Write 50 s product of its prime. 1 Find fctor pir of the given 50 number nd begin the fctor tree (50 = 5 10). 5 10 2 If brnch
More information5.6 The Law of Cosines
44 HPTER 5 nlyti Trigonometry 5.6 The Lw of osines Wht you ll lern out Deriving the Lw of osines Solving Tringles (SS, SSS) Tringle re nd Heron s Formul pplitions... nd why The Lw of osines is n importnt
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationBasic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }
ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All
More informationContent Objectives: After completing the activity, students will gain experience of informally proving Pythagoras Theorem
Pythgors Theorem S Topic 1 Level: Key Stge 3 Dimension: Mesures, Shpe nd Spce Module: Lerning Geometry through Deductive Approch Unit: Pythgors Theorem Student ility: Averge Content Ojectives: After completing
More informationRight Triangle Trigonometry for College Algebra
Right Tringle Trigonometry for ollege Alger B A sin os A = = djent A = = tn A = = djent sin B = = djent os B = = tn B = = djent ontents I. Bkground nd Definitions (exerises on pges 34) II. The Trigonometri
More informationProblem Set 2 Solutions
University of Cliforni, Berkeley Spring 2012 EE 42/100 Prof. A. Niknej Prolem Set 2 Solutions Plese note tht these re merely suggeste solutions. Mny of these prolems n e pprohe in ifferent wys. 1. In prolems
More informationSolutions to Section 1
Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this
More informationAngles and Triangles
nges nd Tringes n nge is formed when two rys hve ommon strting point or vertex. The mesure of n nge is given in degrees, with ompete revoution representing 360 degrees. Some fmiir nges inude nother fmiir
More informationIt may be helpful to review some right triangle trigonometry. Given the right triangle: C = 90º
Ryn Lenet Pge 1 Chemistry 511 Experiment: The Hydrogen Emission Spetrum Introdution When we view white light through diffrtion grting, we n see ll of the omponents of the visible spetr. (ROYGBIV) The diffrtion
More informationQuadratic Equations  1
Alger Module A60 Qudrtic Equtions  1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions  1 Sttement of Prerequisite
More informationSection 55 Solving Right Triangles*
55 Solving Right Tringles 379 79. Geometry. The re of retngulr nsided polygon irumsried out irle of rdius is given y A n tn 80 n (A) Find A for n 8, n 00, n,000, nd n 0,000. Compute eh to five deiml
More information10.5 Graphing Quadratic Functions
0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions
More informationChapter 9: Quadratic Equations
Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.
More informationHow to Graphically Interpret the Complex Roots of a Quadratic Equation
Universit of Nersk  Linoln DigitlCommons@Universit of Nersk  Linoln MAT Em Epositor Ppers Mth in the Middle Institute Prtnership 7007 How to Grphill Interpret the Comple Roots of Qudrti Eqution Crmen
More informationConsumption and investment spending. Cambridge University Press 2012 Economics for the IB Diploma 1
Supplementry mteril for Chpter 9 9.8 Understnding ggregte demnd nd the multiplier in terms of the Keynesin ross model (supplementry mteril, reommended for higher level) This mteril is inluded for the interested
More informationCHAPTER 4: POLYGONS AND SOLIDS. 3 Which of the following are regular polygons? 4 Draw a pentagon with equal sides but with unequal angles.
Mthemtis for Austrli Yer 6  Homework POLYGONS AND SOLIDS (Chpter 4) CHAPTER 4: POLYGONS AND SOLIDS 4A POLYGONS 3 Whih of the following re regulr polygons? A polygon is lose figure whih hs only stright
More informationPythagoras theorem is one of the most popular theorems. Paper Folding And The Theorem of Pythagoras. Visual Connect in Teaching.
in the lssroom Visul Connet in Tehing Pper Folding And The Theorem of Pythgors Cn unfolding pper ot revel proof of Pythgors theorem? Does mking squre within squre e nything more thn n exerise in geometry
More information8. Hyperbolic triangles
8. Hyperoli tringles Note: This yer, I m not doing this mteril, prt from Pythgors theorem, in the letures (nd, s suh, the reminder isn t exminle). I ve left the mteril s Leture 8 so tht (i) nyody interested
More informationGRADE 4. Fractions WORKSHEETS
GRADE Frtions WORKSHEETS Types of frtions equivlent frtions This frtion wll shows frtions tht re equivlent. Equivlent frtions re frtions tht re the sme mount. How mny equivlent frtions n you fin? Lel eh
More informationEXAMPLE EXAMPLE. Quick Check EXAMPLE EXAMPLE. Quick Check. EXAMPLE RealWorld Connection EXAMPLE
 Wht You ll Lern To use the Pthgoren Theorem To use the onverse of the Pthgoren Theorem... nd Wh To find the distne etween two doks on lke, s in Emple The Pthgoren Theorem nd Its onverse hek Skills You
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationNotes for Thurs 8 Sept Calculus II Fall 2005 New York University Instructor: Tyler Neylon Scribe: Kelsey Williams
Notes for Thurs 8 Sept Clculus II Fll 00 New York University Instructor: Tyler Neylon Scribe: Kelsey Willims 8. Integrtion by Prts This section is primrily bout the formul u dv = uv v ( ) which is essentilly
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More information. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2
7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse rel number to its binry representtion,. convert binry number to n equivlent bse number. In everydy
More informationAnswer, Key Homework 10 David McIntyre 1
Answer, Key Homework 10 Dvid McIntyre 1 This printout should hve 22 questions, check tht it is complete. Multiplechoice questions my continue on the next column or pge: find ll choices efore mking your
More informationPythagoras theorem and trigonometry (2)
HPTR 10 Pythgors theorem nd trigonometry (2) 31 HPTR Liner equtions In hpter 19, Pythgors theorem nd trigonometry were used to find the lengths of sides nd the sizes of ngles in rightngled tringles. These
More informationPYTHAGORAS THEOREM. Answers. Edexcel GCSE Mathematics (Linear) 1MA0
Edexel GSE Mthemtis (Liner) 1M0 nswers PYTHGORS THEOREM Mterils required for exmintion Ruler grduted in entimetres nd millimetres, protrtor, ompsses, pen, H penil, erser. Tring pper my e used. Items inluded
More information1 Fractions from an advanced point of view
1 Frtions from n vne point of view We re going to stuy frtions from the viewpoint of moern lger, or strt lger. Our gol is to evelop eeper unerstning of wht n men. One onsequene of our eeper unerstning
More informationThe art of Paperarchitecture (PA). MANUAL
The rt of Pperrhiteture (PA). MANUAL Introution Pperrhiteture (PA) is the rt of reting threeimensionl (3D) ojets out of plin piee of pper or ror. At first, esign is rwn (mnully or printe (using grphil
More informationASYMPTOTES HORIZONTAL ASYMPTOTES VERTICAL ASYMPTOTES. An asymptote is a line which a function gets closer and closer to but never quite reaches.
UNFAMILIAR FUNCTIONS (Chpter 19) 547 B ASYMPTOTES An smptote is line whih funtion gets loser n loser to but never quite rehes. In this ourse we onsier smptotes whih re horizontl or vertil. HORIZONTAL ASYMPTOTES
More informationRightangled triangles
13 13A Pythgors theorem 13B Clulting trigonometri rtios 13C Finding n unknown side 13D Finding ngles 13E Angles of elevtion nd depression Rightngled tringles Syllus referene Mesurement 4 Rightngled tringles
More informationThe Cat in the Hat. by Dr. Seuss. A a. B b. A a. Rich Vocabulary. Learning Ab Rhyming
MINILESSON IN TION The t in the Ht y Dr. Seuss Rih Voulry tme dj. esy to hndle (not wild) LERNING Lerning Rhyming OUT Words I know it is wet nd the sun is not sunny. ut we n hve Lots of good fun tht is
More information