Q u a d r a t i c E q u a t i o n s QUADRATIC EQUATIONS.

Transcription

1 Q u d r t i E q u t i o n s QUADRATIC EQUATIONS

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3 Qudrti QUADRATIC Equtions EQUATIONS A qudrti eqution is n eqution where the highest inde is (squred). These re little me omplited to solve thn liner equtions, nd my hve no rel solutions t ll. Answer these questions, efe wking through the hpter. I used to think: How mny solutions does qudrti eqution hve? Is it possile to find solutions f in + + = sed only on, nd? If = 16, is = 4 the only nswer? Answer these questions, fter wking through the hpter. But now I think: How mny solutions does qudrti eqution hve? Is it possile to find solutions f in + + = sed only on, nd? If = 16, is = 4 the only nswer? Wht do I know now tht I didn't know efe? 1% Qudrti Equtions Mthletis 1% P Lerning SERIES TOPIC 1

4 Qudrti Equtions Bsis Qudrti Equtions If the highest inde of the vrile is (eg. y ) then the eqution is not liner, ut is lled qudrti eqution. To find the vrile y itself on one side, the squre root is used to hnge to. But the squre root must e used on oth sides. Solve f in the qudrti eqution + 16 = = 4-16 = 18 = 9 Find the squre root of oth sides = ^! h =! Sutrt 16 from oth sides Divide oth sides y Beuse = 9 nd ^- h = 9 Eh qudrti eqution n hve one, two no rel solutions. The ove emple hs two solutions ( = - = ). Sometimes the solutions will hve to e left in surd fm like this net emple: Solve f 4 - = 4 = = 5 = ^! 5 h =! 5 This eqution hs the two possile solutions = 5 = 5 Here is n emple of qudrti eqution with no rel solutions: Solve f in the qudrti eqution + 8 = =-4-8 =! -4 This does not hve ny rel solutions sine no rel numer is the squre root of negtive numer. = - 1 =-4 1% Qudrti Equtions SERIES TOPIC Mthletis 1% P Lerning

5 Qudrti Equtions Bsis 1. Wht is the differene etween liner nd qudrti eqution?. Find the solutions to the vrile in eh of these equtions. Leve the solutions in surd fm if neessry, nd stte if eh eqution hs one, two no rel solutions: = 4 = - 75 = d 6y = e 4t - 8 = f h =-9 g p =-8 h 8m + 5 = 5 i 9-16 = j - k + 18 = 1% Qudrti Equtions Mthletis 1% P Lerning SERIES TOPIC

6 Qudrti Equtions Knowing Me The Null Ft Lw Let's sy there re two numers nd. If # = then = = ( oth re ) This mens tht if the produt of two epressions is zero, then t lest one of the epressions must e equl to zero. Here re some emples of how the null ft lw is used to solve equtions: Solve f ^ + h= ^-1h^- h= ` # ^+ h= ` ^-1h# ^- h= ` = + = ^nullft lwh ` - 1 = - = ^nullftlwh ` = =- ` = 1 = Ftising to Solve Qudrti Equtions Type 1: + = Ftise to 1 rket nd then use the null ft lw to find two possile solutions. Solve f in these qudrti equtions + 4 = - 15 = + 4 = - 15 = ^ + 4h = ^- 15h = # ^+ 4h = # ^- 15h = ` = + 4 = ` = - 15 = ` = =-4 ` = = 5 Type : + + = Ftise to rkets nd then use the null ft lw to find two possile solutions. Solve f in these qudrti equtions + = = = ` ^- 1h^+ 5h= ` ^- h^+ 4h= ` - 1 = + 5 = ` - = + 4 = ` = =-4 ` = 1 =-5 ` = 1 =-5 4 1% Qudrti Equtions SERIES TOPIC Mthletis 1% P Lerning

7 Qudrti Equtions Knowing Me 1. Rewrite these polynomils in the fm + + =. Identify the vlues of, nd : + + = = - 7 = d ^ + 4h= e ^4-5h= f ^+ h^- 7h= g ^+ 5h^- 8h= h - ^+ 4h^- 1h=. Solve these qudrti equtions f the missing vrile: ^ - h= ^ + 4h= ^+ 1h= d ^- h^+ 5h= e y^y- h= f ^+ 7h^- 8h= 1% Qudrti Equtions Mthletis 1% P Lerning SERIES TOPIC 5

8 Qudrti Equtions Knowing Me. Solve these qudrti equtions y ftising: + 8 = = t - 6t = d 5-15 = e y -4y- 1 = f m -m- = 6 1% Qudrti Equtions SERIES TOPIC Mthletis 1% P Lerning

9 Qudrti Equtions Knowing Me g n + 8n- 64 = h + 5 = 6 i ^ + h= 15 j ^p+ h = 8p+ 7 1% Qudrti Equtions Mthletis 1% P Lerning SERIES TOPIC 7

10 Qudrti Equtions Knowing Me Completing the Squre The trinomil n't e ftised into two rkets s esily s the emples in the previous setion. So to solve = the trinomil on the left needs to e trnsfmed different wy. This is lled Completing the Squre is used in n emple to show how to omplete the squre elow: Complete the squre of Step 1: Write rket squred with plus hlf of the oeffiient of = Hlf = ^ + h Sutrt Step : Sutrt the squre of the numer in the rket = ^ + h - 9 Squred Step : Write in the onstnt term from the trinomil Step 4: Simplify ` = ^ + h - 8 = ^ + h = ^ + h - 8 In the ove emple the oeffiient of is 1. If it is not 1 then it must e fted out first. Complete the squre of Step 1: Ft out the oeffiient of using squre rkets = = 46-4 Hlf Step : Inside the squre rket write round rket squred with plus hlf the numer in front of in the squre rket Step : Inside the squre rket sutrt the squre of the numer in the round rket = 46 ^ - Sutrt = 46 ^ - h - 4@ Squred Step 4: Write the onstnt term in the squre rket from Step 1 ` = 4^ -h - 4 = 46^ -h -4 = 46^ -h - 6@ = 4^ -h % Qudrti Equtions SERIES TOPIC Mthletis 1% P Lerning

11 Qudrti Equtions Knowing Me 4. Complete the squre of the following trinomils: d Hint: 9 ` j = 4 e f Hint: ftise -1 out using squre rkets 1% Qudrti Equtions Mthletis 1% P Lerning SERIES TOPIC 9

12 Qudrti Equtions Knowing Me Why is Completing the Squre Imptnt? If trinomil in n eqution n't e ftised esily into two rkets, then ompleting the squre n e used to solve the eqution. Solve f if = n't e ftised esily into two rkets. Completing the squre n e used to show: = ^h -8 So the iginl eqution n e rewritten s: ^ -h - 8 = Simplify so tht the squred ket is y itself on the left hnd side: ^ -h = + 8 ^ - h = 8 Solve f the rket y finding the squre root of oth sides: ^ - h = ^! 8h ` - =! 8 Solve f the two possile solutions of : - = 8 - =- 8 ` = + 8 = - 8 Solutions re usully left in surd fm, unless question speifies otherwise. Solve the eqution: = Using ompleting the squre: = 4( -) - 4 Rewrite the eqution in the ompleting the squre fm: 4^ -h - 4 = Simplify so tht the squred rket is y itself on the left hnd side: 4^ - h = 4 ^ - h = 6 Solve f the rket y finding the squre root of oth sides: ^ - h = ^! 6h ` - =! 6 Solve f the two possile solutions of : - = 6 - =- 6 ` = + 6 = % Qudrti Equtions SERIES TOPIC Mthletis 1% P Lerning

13 Qudrti Equtions Knowing Me 5. Solve f in the following: ^ + h = 5 ^ - h = 16 ^ - 7h = 8 d - 5^ + 6h = 5 6. Complete the squre nd then solve f in the following: = = 1% Qudrti Equtions Mthletis 1% P Lerning SERIES TOPIC 11

14 Qudrti Equtions Knowing Me 7. Complete the squre nd solve f the vrile in the following: q + q - = -m - m+ 5 = 4t + 8t - 1 = d = 1 1% Qudrti Equtions SERIES TOPIC Mthletis 1% P Lerning

15 Qudrti Equtions Knowing Me 8. Consider the eqution + + = where, nd re ny onstnts: Complete the squre f this eqution. Solve f. 1% Qudrti Equtions Mthletis 1% P Lerning SERIES TOPIC 1

16 Qudrti Equtions Using Our Knowledge The Qudrti Fmul 'The Qudrti Fmul' is lso know s 'The Fmul' f qudrti equtions. It is fmul tht solves f ny qudrti eqution of the fm + + = y using only the oeffiients, nd. The fmul is found from solving + + = using ompleting the squre. ` + + = + + = ` ` + j - + = 4 ` 4 ` + j = - 4 ` + =! - 4 ` = -! 4 - ` = = This mens tht solution n e found f qudrti eqution just using the oeffiients. Solve the following equtions + 5+ = = 1, = 5, = Qudrti fmul Sustitute, nd into qudrti fmul = -! 4-5! 5 41 = - - ^ h^ h 1 ^ h = - 5! 1 = =- 5-1 Solutions re left in surd fm unless question sys otherwise. 14 1% Qudrti Equtions SERIES TOPIC Mthletis 1% P Lerning

17 Qudrti Equtions Using Our Knowledge 1. Use the qudrti fmul to solve the following qudrti equtions, leving your nswers in surd fm: = = = d 4 - t + t- 6 = e y + y- 5 = f -p - 5p+ 5 = 1% Qudrti Equtions Mthletis 1% P Lerning SERIES TOPIC 15

18 Qudrti Equtions Using Our Knowledge. Use the qudrti fmul to solve the following qudrti equtions, leving your nswers in surd fm: = 4-9 = ^- 7h=- 4 d ^4-5h= % Qudrti Equtions SERIES TOPIC Mthletis 1% P Lerning

19 Qudrti Equtions Using Our Knowledge. Solve = using eh of the following methods. (You should get the sme solutions eh time) Ftise nd solve f. Use ompleting the squre to solve f. Use the qudrti fmul to solve f. 1% Qudrti Equtions 17 Mthletis 1% P Lerning SERIES TOPIC

20 Qudrti Equtions Using Our Knowledge The Disriminnt Thnks to the qudrti fmul we know tht solutions to qudrti eqution + + = n e found using: = -! 4 - Notie tht if - 4 is negtive then the surd n't e simplified - sine there is no rel numer whih is the squre root of negtive numer. So, if then there re no rel solutions f. The epression - 4 (the vlue under the surd in the qudrti fmul) is lled the disriminnt, nd represented y T ('Delt'). The disriminnt predits the properties of solutions to qudrti equtions: If T = (negtive) then there will e no rel solutions f. If T = - 4 = then the two solutions will e equl. These equl solutions will lso e rtionl. If T = - 4 (positive) nd is perfet squre then there re two solutions whih re unequl nd rtionl. If T = - 4 (positive) nd is not perfet squre, then there re two solutions whih re unequl nd irrtionl. Hve look t these emples = T = - 4 = ^1h -4 ^ h^- 7h = 57 ` T nd is not perfet squre ` the solutions will e unequl nd irrtionl Using the qudrti fmul: = - 1! 57 4 ` = = Solutions re unequl nd irrtionl (As epeted sine T nd not perfet squre) = T = - 4 = ^1h - 4^1h^5h = ` T = ` the solutions will e equl Using the qudrti fmul: ` =- 5+ =-5- ` =- 5 solutions re equl = - 1! Only one solution (s epeted, sine T = ) + + = T = - 4 = ^h - 4 ^ h^h =-15 ` T1 ` there re no rel solutions f. Using the qudrti fmul: ` the surd n't e simplified ` no rel solutions f (As epeted, sine T1 ) = -! % Qudrti Equtions SERIES TOPIC Mthletis 1% P Lerning

21 Qudrti Equtions Using Our Knowledge The disriminnt n lso e used to determine oeffiients. Find vlues of k so tht - 6+ k = hs: Equl solutions No rel solution F solutions T = - 4 = F no rel solutions T = -4 1 ` ^-6h - 4^h^kh = ` 6-1k = ` 1k = 6 ` k = ` if k = then the eqution will hve equl solutions ` ^-6h -4^h^kh 1 ` 6-1k 1 ` 1k 6 ` k ` if k is greter thn then the eqution will hve no rel solutions Two rel nd unequl solutions F rel nd unequl solutions = -4 ` ^-6h -4^h^kh ` 6-1k ` 1k 1 6 ` k 1 ` if k is less thn then the eqution will hve rel nd unequl solutions m - 5 =- 6 hs equl solutions. Find m First, write the eqution in stndrd fm: m = ` = m, =- 5nd = 6 F equl roots T = - 4 = ` ^-5h - 4^mh^6h = 5-4m = ` m = 5 4 1% Qudrti Equtions Mthletis 1% P Lerning SERIES TOPIC 19

22 Qudrti Equtions Using Our Knowledge 4. Use the disriminnt to determine if these equtions hve two equl, unequl no rel solutions. (Hint: Mke sure the eqution is in stndrd fm + + = ) = = = d - 18 = 16 e = -4 f 8-8 = 1% Qudrti Equtions SERIES TOPIC Mthletis 1% P Lerning

23 Qudrti Equtions Using Our Knowledge 5. Find the vlue of r if r = : Hs one solution. Hs two unequl solutions. 6. Find the vlues f k suh tht + + 7k = 6k- hs no rel solutions: 1% Qudrti Equtions Mthletis 1% P Lerning SERIES TOPIC 1

24 Qudrti Equtions Using Our Knowledge 7. Use the disriminnt to prove tht r + r + r = hs equl solutions f ny r: 8. Prove tht + k+ k = k - k never hs rel roots f ny k: 9. Find p if the following eqution hs two unequl rel solutions: = - p 5-4 1% Qudrti Equtions SERIES TOPIC Mthletis 1% P Lerning

25 Qudrti Equtions Thinking Me Equtions tht Redue to Qudrtis Some equtions n look very omplited, ut they n e mde esier y mking vrile sustitution to trnsfm nonqudrti eqution into qudrti eqution. This is done in esy steps: Step 1: Mke sustitution to redue the omplited eqution to qudrti eqution Step : Solve the qudrti eqution Step : Sustitute the iginl vrile k nd solve F emple, solve f if ^ - 7h + ^-7h- = Step 1: sustitute k = 7 to trnsfm the omplited eqution to k + k- = Step : this is qudrti eqution whih n esily e ftised ` ^k- 5h^k+ 4h= ` k - 5 = k = 5 k + 4 = k =-4 Step : sustitute the iginl vriles k in ` - 7 = 5 ` = =-4 = 4 Solve f if -1-8 = Step 1: sustitute k = to trnsfm the omplited eqution to k -1k- 8 = ` k -5k- 14 = Step : this is qudrti eqution whih n esily e ftised ` ^k- 7h^k+ h= ` k - 7 = k = 7 k + = k =- Step : sustitute the iginl vriles k in = 7 ` =! 7 ` = 7 =- 7 =- =! - This is impossile 1% Qudrti Equtions Mthletis 1% P Lerning SERIES TOPIC

26 Qudrti Equtions Thinking Me 1. Sustitute k = 6 to solve f in =.. Mke sustitution to solve f in: 4 + 6^7+ h- ^7+ h = ^-h 4-5^-h - 6 = 9-1^ h+ 9 = ^Hint:9 = h 4 1% Qudrti Equtions SERIES TOPIC Mthletis 1% P Lerning

27 Qudrti Equtions Thinking Even Me Qudrti Wd Prolems Some prolems n e represented y qudrti eqution. Any of the methods in this hpter n e used to solve these prolems. When hoosing method to use, most people find it esier to try ftising into two rkets first. If n eqution n't e ftised into two rkets esily, then use the qudrti fmul ompleting the squre. Use the Pythgen Theem to find the lengths of the missing sides in the following right ngled tringle: 5m Ading to the Pythgen theem: This n e ftised into two rkets + ^+ 1h = 5 ` = ` = ^- h^+ 4h= ` = = The length of the side of tringle n't e negtive, so = is the only solution. The missing sides of the tringle re = m nd + 1 = 4m Here is n emple of wd prolem whih n e represented y qudrti eqution. To find n eqution from wd prolem, let vrile equl the missing vlue nd use the infmtion in the wd prolem to rete n eqution. Sndr is twie s old s Russell. Eight yers go, the produt of their ges ws 1. Find their urrent ges: Let Russell's ge e ` Sndr's ge is Eight yers go their ges were: Russell: - 8 Sndr: - 8 From the wd prolem we n sy ^-8h^- 8h= 1 ` = = ` = This n e ftised into two rkets ^-h^- 9h= ` = = 9 It is impossile f Russell to e yers old sine he would not yet hve een n 8 yers go. ` = 9 is the only solution ` Russell is urrently 9 yers nd Sndr is urrently ^9 h = 18 yers old. 1% Qudrti Equtions Mthletis 1% P Lerning SERIES TOPIC 5

28 Qudrti Equtions Thinking Even Me 1. The produt of two onseutive integers is 7. Find the two numers if they re positive. Find the two numers if they re negtive.. A retngle s length is less thn four times the redth. Find the dimensions of the retngle if the re is 16m : 6 1% Qudrti Equtions SERIES TOPIC Mthletis 1% P Lerning

29 Qudrti Equtions Thinking Even Me. The stge elow is mde up of squre nd retngle. Find If the totl re of the stge is 191m % Qudrti Equtions Mthletis 1% P Lerning SERIES TOPIC 7

30 Qudrti Equtions Thinking Even Me Simultneous Equtions with Qudrtis In the sme wy simultneous equtions eist f liner equtions, they eist f qudrti equtions too. When you re solving simultneous equtions involving liner eqution nd qudrti eqution lwys follow these four steps: Step 1: Mke y the sujet in oth equtions Step : Sustitute the qudrti eqution into the liner eqution Step : Solve f Step 4: Sustitute the -vlue into the iginl liner eqution ( the iginl qudrti eqution) to solve f y Solve f these simultneous equtions y- 1 = - 7 y- = Step 1: Mke y the sujet of oth equtions y = y = + Step : Sustitute the qudrti eqution into the liner eqution Sustitute into 1 ^ h= + Step : Solve f = = ^-1h^- 7h= ^14, h ^71, h ` = 1 = 7 Step 4: Sustitute the -vlue into the iginl liner eqution ( the iginl qudrti eqution) to solve f y y = ^1h+ y = 4 y = ^7h+ y = 1 ` = 1 nd y = 4 = 7 nd y = 1 This mens tht the grphs of the equtions will interset t (1, 4) nd (7, 1) 8 1% Qudrti Equtions SERIES TOPIC Mthletis 1% P Lerning

31 Qudrti Equtions Thinking Even Me A stright line (liner eqution) nd prol (qudrti eqution) ould interset t one point, two points not t ll. If the qudrti eqution in step hs two solutions, then the stright line nd prol interset t two points nd so the stright line is sent. If the eqution in step hs one solution (the solutions re equl) then the stright line touhes the prol only one nd so the stright line is tngent to the prol. If the eqution in step hs no rel solutions ^ - 1 h then the stright line nd prol will not interset t ll. 1% Qudrti Equtions Mthletis 1% P Lerning SERIES TOPIC 9

32 Qudrti Equtions Thinking Even Me 4. Solve these simultneous equtions f nd y. How mny times does the stright line interset the prol? y = y = % Qudrti Equtions SERIES TOPIC Mthletis 1% P Lerning

33 Qudrti Equtions Thinking Even Me 5. Solve these simultneous equtions f nd y. How mny times does the stright line interset the prol? y- 5 = 4-1 y+ = 1% Qudrti Equtions Mthletis 1% P Lerning SERIES TOPIC 1

34 Qudrti Equtions Thinking Even Me 6. Solve these simultneous equtions f nd y. Find the odintes of the point(s) where the grphs would interset: y- = = 7- y 1% Qudrti Equtions SERIES TOPIC Mthletis 1% P Lerning

35 Qudrti Equtions Thinking Even Me 7. Solve these simultneous equtions f nd y. Find the odintes of the point(s) where the grphs would interset: y = y = % Qudrti Equtions Mthletis 1% P Lerning SERIES TOPIC

36 Qudrti Equtions Answers Bsis: Knowing Me: 1. In liner eqution the highest inde of the vrile (eg ) hs n inde of 1. In qudrti eqution the highest inde of the vrile hs n inde of (eg ).. = = = - = = 4 + = (Null Ft Lw) (Null Ft Lw). =! =! 4 Two solutions Two solutions =! 5 Two solutions d y = One solution = = = 4 =- + 1 = =- 1 (Null Ft Lw) e t =! 7 Two solutions f h =-9 No rel solutions d - = = 5 + = =-5 (Null Ft Lw) g p 4 h m = No rel solutions One solution i =! 4 j k =! 6 Two solutions Two solutions Knowing Me: e f y = y = + 7 = =- 7 y - = y = - 8 = = 8 (Null Ft Lw) (Null Ft Lw) = ` = 1, =, =. = 8 + = (Null Ft Lw) = ` =, = 4, = = ` = 1, =, =-7 = 4 = = 8 =- + = =- (Null Ft Lw) d = ` = 1, = 4, = t = t = t 6 - = t = 6 (Null Ft Lw) e = ` = 1, =- 15, = d 5 = - = (Null Ft Lw) f g = ` = 1, =- 4, = = ` =, =- 19, =-4 e = y 7 - = y 7 = = y + = y =- (Null Ft Lw) h = ` =-, =- 9, = 1 f m 6 - = m 6 = m 5 + = m 5 =- (Null Ft Lw) 4 1% Qudrti Equtions SERIES TOPIC Mthletis 1% P Lerning

37 Qudrti Equtions Answers Knowing Me:. g n + 8 = n - 4 = (Null Ft Lw) 8. h n 8 =- 1 - = 1 = n 4 = 6 + = 6 =- (Null Ft Lw) Knowing Me: ` + j = - 4 = -! 4 - Using Our Knowledge: i 7 + = 5 - = (Null Ft Lw) 1. = 6 =- 7 =- 5 = = 7 = j p 9 - = p 7 + = (Null Ft Lw) =-8 =-5 p 9 = p 7 =- d = 5 = 4. ^ + h - ^ - h d ` + + j ^ + 4h e f = =-1-6 = =- 5-5 e -^ - 5h + f - ^ + 1h + 1. =-4 = 5. 5 =- + 5 =- - = =- d =-1 = 7+ 7 = No rel solution = 7- d = = = = =- + 7 = ^- h^+ 8h@ = - = 8 + = (Null Ft Lw) 5 1 = = - = 8 =- 7. q = 1 q =- ^ + h = ^! 5h = =-8 m = t = m 1 6 =- - t = = - 4! 7 14 =-8 = d = + 1 = - 1 1% Qudrti Equtions Mthletis 1% P Lerning SERIES TOPIC 5

38 Qudrti Equtions Answers Using Our Knowledge: Thinking Even Me: 4. T nd is perfet squre ` the solutions will e unequl nd rtionl T nd is perfet squre ` the solutions will e unequl nd rtionl T1 ` there re no rel solutions f 1.. The positive numers re 16 nd 17 The negtive numers re -16 nd -17 Bredth = 6m Length is = 1m d T nd is perfet squre ` the solutions will e unequl nd rtionl. = 7m e f T nd is not perfet squre ` the solutions will e unequl nd irrtionl T =. The two solutions will e equl These equl solutions will lso e rtionl 4. = 5 =- y = 1 y =-9 Therefe the stright line intersets the prol twie t: ^1, h nd ^-5, -9h 5. r = hs one solution when r = 1 4 The eqution r = hs unequl solutions when r = y 1 = Therefe the stright line intersets the prol twie t: ^1, h nd ^1, -1h 1 = 1 = y 1 =- 6 = k = 6k- hs no rel solutions when k 4 y 9 = y 19 =- Therefe the grphs interset t: ^1, 9h nd ^-6, -19 h 9. = - hs two unequl nd rel p 5-4 solutions when p 1 1 p The disriminnt is 1 so this n't e solved (no rel solutions) Thinking Me: 1. ` = = -4. ` =-1 = 1 ` =-1 = 5 ` = = 6 1% Qudrti Equtions SERIES TOPIC Mthletis 1% P Lerning

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