Quadratic Equations  1


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1 Alger Module A60 Qudrtic Equtions  1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008
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3 Qudrtic Equtions  1 Sttement of Prerequisite Skills Complete ll previous TLM modules efore completing this module. Required Supporting Mterils Access to the World Wide We. Internet Eplorer 5.5 or greter. Mcromedi Flsh Plyer. Rtionle Why is it importnt for you to lern this mteril? The qudrtic eqution is one of the most common equtions encountered in the technologies. Its pplictions include projectile motion, rchitecturl prolems, electricl circuits, prolems in mechnics, nd design. This unit will introduce the student to mny different strtegies for mnipulting nd solving qudrtic equtions. Lerning Outcome When you complete this module you will e le to Solve qudrtic equtions using vriety of methods. Lerning Ojectives 1. Rewrite lgeric epressions in the stndrd qudrtic form nd stte the vlues of,, nd c.. Solve pure qudrtic equtions.. Solve qudrtic equtions y fctoring.. Solve qudrtic eqution y completing the squre. 5. Solve qudrtic eqution y formul. 6. Solve qudrtic using ny method. Connection Activity Consider the following tringle: Your previous eperience with the Pythgoren theorem tells you tht + ( + 1) 6. Simplifying this eqution we get or This is qudrtic eqution nd solving the eqution will llow you to solve for the length of the sides of the tringle (). Module A60 Qudrtic Equtions  1 1
4 OBJECTIVE ONE When you complete this ojective you will e le to Rewrite lgeric epressions in the stndrd qudrtic form nd stte the vlues of,, nd c. Eplortion Activity Given tht, nd c re constnts nd 0, the eqution: + + c 0 is clled the Stndrd Form of qudrtic eqution. Note: When identifying, nd c, include the signs if they re negtive. EXAMPLES This is in the form of + + c 0 nd 0. Therefore it is qudrtic eqution with,, c Qudrtic: yes or no? Yes, with, 7, c Qudrtic: yes or no? No, this is not qudrtic ecuse of the presence of the term, nd lso ecuse This is not in the generl qudrtic form, ut it cn e rerrnged. Move ll the terms on the right hnd side of the eqution to the left hnd side, y sutrcting or dding. Then get: which simplifies to: This is qudrtic with 1, 16, nd c ( 7) + + This eqution must e epnded nd simplified first. Drop rckets: Now collect like terms to get: Here 1, 11, nd c 0. Module A60 Qudrtic Equtions  1
5 Eperientil Activity One Determine which of the following equtions re qudrtics. If the stndrd form + + c 0 cn e otined, identify, nd c ( 1). ( + 8) 5. ( + ) 0 6. ( 1) + 7. (1 6) 8. (1 + ) 9. (1 ) 10. ( + 1) ( + 1) ( + 6) 1 ( + 1) ( ) 15. ( ) Show Me. Eperientil Activity One Answers 1. 1; 7; c. ; 9; c 5. Not qudrtic ( 0). Not qudrtic ( 0) 5. l; ; c 6. 1; 1; c ; 1; c 0 8. ; ; c 0 9. Not qudrtic 10. Not qudrtic 11. 6; ; c ; 6; c 1 1. l; 6; c 0 1. l; l0; c ; 8; c ; 0; c 8 Module A60 Qudrtic Equtions  1
6 OBJECTIVE TWO When you complete this ojective you will e le to Solve pure qudrtic equtions. Eplortion Activity A qudrtic eqution in which the term is missing, i.e. in which 0, is clled pure qudrtic eqution. EXAMPLES 1. R 9 0 Note: 1, 0, c 9. 1 Note:, 0, c 1 Solving Pure Qudrtic Eqution. EXAMPLE 1 Solve: SOLUTION: Isolte y dding 16 to oth sides Now tke the squre roots of oth sides. 16 nd ± Check: 16 0 i) if ii) if () Module A60 Qudrtic Equtions  1
7 EXAMPLE Solve 5A SOLUTION: Isolte A. Add 89 to oth sides to get: 5A A 180 Now divide y 5, A A 6 Tke the squre root of oth sides to get: Check: A ±6 i) if A 6 5A ii) if A 6 5A (6) Module A60 Qudrtic Equtions  1 5
8 Eperientil Activity Two Solve the following nd check your nswers (m +1 ) m(m ) 5m Show Me. Eperientil Activity Two Answers Note: Answers re not supplied for these eercises in the hope tht the student will get in the hit of CHECKING his/her nswers to see if they re correct. Deciml nswers my not lwys check ectly ecuse of the numer of deciml plces tken. Close in these cses is good enough. Try to void using decimls nd use frctions insted. 6 Module A60 Qudrtic Equtions  1
9 OBJECTIVE THREE When you complete this ojective you will e le to Solve qudrtic equtions y fctoring. Eplortion Activity If the polynomil + + c cn e fctored esily, then there is simple method for solving the eqution + + c 0. The method is sed on the ZeroProduct theorem. ZeroProduct Theorem: If the product of two rel numers, nd, is zero, then one of the fctors of the product, or must e zero. Tht is, 0 if 0 or 0. This theorem cn e etended to lgeric epressions. For instnce if ( + )( ) 0 this is possile if nd only if either ( + ) 0, or ( ) 0 Solving qudrtics y fctoring. EXAMPLE 1 Solve for nd check your nswer. SOLUTION: To fctor we must find integers tht hve product of +, nd sum of +5. nd 1 seem likely numers to try so we hve: ( + )( + 1) 0 Now to solve, we set ech fctor to zero nd solve the eqution. so nd 1 Check: i) if () + 5() Thus () checks. ii) nd for (1) + 5(1) + 0. Thus (1) checks. Module A60 Qudrtic Equtions  1 7
10 EXAMPLE Solve nd check. SOLUTION: Arrnge the eqution into proper formt Fctor: use tril nd error method. ( 1)( + 5) 0 Set ech fctor equl to zero nd solve. 1 0 nd nd Check: i) if ( 1 ) 5 7( 1 ) 5 ii) if ( ) 5 5 7( ) HINT: When you re trying to find the fctors of qudrtic such s the previous emple where the coefficient of the term hs severl comintions of fctors, i.e nd 6, try the comintion in which the fctors re close in size, i.e. try the nd. Tking the etremes, i.e. 6 nd 1 my work ut more often the "closer" fctors led you into the solution of the prolem quicker. This ide lso works when looking t the constnt term. For instnce if c, do not choose nd 1 s fctors, ut strt with 6 nd nd work from there. (Good luck, s there is no esy wy out. Prctice, prctice, prctice. ) Also, look for the ptterns in numers, lwys keeping in mind the middle term. 8 Module A60 Qudrtic Equtions  1
11 Eperientil Activity Three Solve the following qudrtics y fctoring nd check your nswers t + t y 8 10y n n 1. v v p + 0 7p Show Me m + m t t 6. r 1r 7. 8m + m ( is constnt) ( is constnt) Eperientil Activity Three Answers Answers re not supplied for these eercises in the hope tht the student will get in the hit of CHECKING his/her nswers to see if they re correct. Deciml nswers my not lwys check ectly ecuse of the numer of deciml plces tken. Close in these cses is good enough. Try to void using decimls nd use frctions insted. Module A60 Qudrtic Equtions  1 9
12 OBJECTIVE FOUR When you complete this ojective you will e le to Solve qudrtic eqution y completing the squre. Eplortion Activity Some qudrtics re not redily solved y fctoring, ut cn e solved y method known s completing the squre. To solve y completing the squre: Step 1: mke sure the coefficient of the term is +1. If it isn't +1 then divide ech term of the eqution y numer tht will mke the coefficient of the term +1. Step : move the c term to the right hnd side. Step : dd constnt to oth sides which mkes the left hnd side perfect squre. This constnt is the squre of 1/ of the coefficient of the term. Step : fctor the left hnd side, s it now perfect squre. Step 5: tke the squre root of oth sides. Step 6: solve for. NOTE: Follow Emple 1 nd keep these steps in mind. EXAMPLE 1 Solve l0 0 0 y completing the squre. SOLUTION: To complete the squre: Check to see if the coefficient of 1. It is, proceed, Move the constnt 0 to the other side of the eqution y ddition 10 0 Now tke 1 the coefficient of the term which is 10, nd squre it, getting 5. Add this 5 to oth sides: Module A60 Qudrtic Equtions  1
13 Fctor the left side s perfect squre: ( 5)( 5) 5 ( 5) 5 Now the tke squre root of oth sides: 5 ± 5 5 ± 5 +5 ± nd 1.71 Ect form Approimte form Check: i) if (11.71) 10(11.71) ii) if 1.71 (1.71) 10( 1.71) 0 0 EXAMPLE Solve 1 0 y completing the squre. SOLUTION: To complete the squre: Check to see if the coefficient of 1. It isn't, so divide through y. 1 1 We get: Now the constnt to e dded to oth sides is otined y tking of squring it: 6 6 Module A60 Qudrtic Equtions
14 So we get: Now tke the squre root of oth sides 1 1 ± ± 6 6 1± nd 0. Check: i) if ii) if Module A60 Qudrtic Equtions  1
15 Eperientil Activity Four Solve y completing the squre nd check your nswers E 15E i 7i θ + θ 8. e 6 e 9. M M E E θ θ 1. 17I I + I φ φ f + f Show Me. 7( R ) R Z 1 Z 15. ( R ) R Z + 1 Z + Eperientil Activity Four Answers Answers re not supplied for these eercises in the hope tht the student will get in the hit of CHECKING his/her nswers to see if they re correct. Deciml nswers my not lwys check ectly ecuse of the numer of deciml plces tken. Close in these cses is good enough. Module A60 Qudrtic Equtions  1 1
16 Module A60 Qudrtic Equtions  1 OBJECTIVE FIVE When you complete this ojective you will e le to Solve qudrtic eqution y formul. Eplortion Activity The formul for solving qudrtic eqution is found y completing the squre for the generl qudrtic eqution. Study the following nd e prepred to develop this formul on supervised test. + + c 0 To Complete the Squre c Coefficient of must e one, therefore divide ech term y. c + Move constnt to the right. c To find the constnt to e dded to oth sides: of 1 tke nd squre it to get c + + Fctor left hnd side. c + c ± + Tke the squre root of oth sides; then solve for. c ± c ± Simplify lgericlly. c c c ± ± ± 1
17 The formul: ± c is known s the qudrtic formul. It is etremely useful! Insted of ttempting to solve qudrtic eqution y fctoring or completing the squre we now cn use the qudrtic formul. EXAMPLE 1 Solve using the qudrtic formul. SOLUTION: Identify,, nd c: 5,, c Now sustitute into the formul: Check: ± ± ± ± 6 10 ± c ( )( 5)( ) ( )( 5) 8 nd 10 so: 0.6 nd 1 i) if ii) if Module A60 Qudrtic Equtions
18 EXAMPLE Using the qudrtic formul, solve for R in R 5 R SOLUTION: Arrnge the eqution in stndrd qudrtic form. Multiply oth sides y (5 R), 5 R R ( 5 R) R( 5 R) ( 5 R) Epnd nd collect like terms, l0r R R l0r + 0 Identify,, nd c., l0, nd c. ± Use R c ( 10) ± ( 10) ( )( ) ( ) 10 ± ± R nd R R.68; nd R Module A60 Qudrtic Equtions  1
19 Check: i) if R.68 R 5 R ii) if R 0.0 R 5 R Module A60 Qudrtic Equtions
20 Eperientil Activity Five Solve the following equtions y using the qudrtic formul nd check your nswers. 1. θ θ I + 5 I q 8q 6. I 7I Z Z 8. 5(R + ) R(R 1) m m I1 I1 I1 1 R R1 R I Show Me. ( ) I + I i 16 7i i + 1 E 5 E E 7E Eperientil Activity Five Answers Answers re not supplied for these eercises in the hope tht the student will get in the hit of CHECKING his/her nswers to see if they re correct. Deciml nswers my not lwys check ectly ecuse of the numer of deciml plces tken. Close in these cses is good enough. 18 Module A60 Qudrtic Equtions  1
21 OBJECTIVE SIX When you complete this ojective you will e le to Solve qudrtic using ny method. Eplortion Activity Solve qudrtic y ny method. It is importnt tht the student e le to use ny of the methods discussed for solving qudrtic eqution. Eperientil Activity Si For ech of the following qudrtic equtions solve y: 1. Fctoring (if possile). Completing the squre. Qudrtic formul Check your nswers y 5y s s + 8. n n 6 9. t 8t Eperientil Activity Si Answers Answers re not supplied for these eercises in the hope tht the student will get in the hit of CHECKING his/her nswers to see if they re correct. Deciml nswers my not lwys check ectly ecuse of the numer of deciml plces tken. Close in these cses is good enough. Prcticl Appliction Activity Complete the Qudrtic Equtions  1 ssignment in TLM. Summry This module introduced vriety of methods for solving qudrtic equtions. Module A60 Qudrtic Equtions
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Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
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