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  • T TH THUT GII TON PHNG TRNH V T - ON TR DNG

    1

    TI LIU N THI

    TRUNG HC PH THNG QUC GIA

    ------------------------***------------------------

    TH THUT

    Gii ton

    PHNG TRNH V T

    Tc gi: ON TR DNG

    H NI, THNG 4 NM 2016

  • TH THUT GII TON PHNG TRNH V T - ON TR DNG

    2

    CH 1: 4 K NNG C BN CN BIT

    TRONG QU TRNH GII TON BNG MY TNH CASIO

    I. K nng 1: K nng nng ly tha:

    K nng nng ly tha l rt quan trng trong qu trnh gii ton

    m trong qu trnh gii ton, ta vn thng gi vi nhng tn quen

    thuc nh bnh phng hai v, lp phng hai v. Hc sinh cn

    nm vng cc hng ng thc c bn v nng ly tha nh sau:

    2 2 2a b a b 2ab .

    3 3 2 2 3a b a 3a b 3ab b .

    2 2 2 2a b c a b c 2 ab bc ca .

    3 3 3 3a b c a b c 3 a b b c c a .

    3 3 3 3a b c a b c 3 a b c ab bc ca 3abc .

    II. K nng 2: Phn tch nhn t biu thc cha mt cn dng c

    bn:

    V d 1: Phn tch nhn t: x 2 x 3

    t 3x 3 t x t 3 . Khi :

    2x 2 x 3 t 2t 3 t 1 t 3 .

    Do thay ngc t x 3 ta c:

    x 2 x 3 x 3 1 x 3 3 . BI TP T LUYN

    Bi 1: Phn tch nhn t: 2x 4 5 x 1

    p n: 2 x 1 1 x 1 2 Bi 2: Phn tch nhn t: 2x 5 7 2x 1

    p n: 2x 1 1 2x 1 6 III. K nng 3: Phn tch nhn t hai bin khng cha cn:

    V d 2: Phn tch nhn t: 2 2x 2xy y x y (Ti a l bc 2).

    Thay y 100 , biu thc tr thnh: 2 2 2x 2xy y x y x 201x 10100 .

    Bm my phng trnh bc 2 ta c 2 nghim: x 100,x 101 .

  • T TH THUT GII TON PHNG TRNH V T - ON TR DNG

    3

    Do : 2x 201x 10100 x 100 x 101 . V 100 y,101 100 1 y 1 , vy:

    2 2x 2xy y x y x y x y 1 .

    V d 3: Phn tch nhn t: 3 2 2 2x 2x y xy y xy 3x 3y .

    Thay y 100 , biu thc tr thnh: 3 2 2 2 3 2x 2x y xy y xy 3x 3y x 200x 10103x 10300

    S dng SOLVE ta c x 100 y . Ta c hai cch x l sau:

    Cch 1: S dng CALC:

    Thay 1

    x 1000,y100

    ta c:

    3 2 2 2x 2x y xy y xy 3x 3y1000013.01

    x y

    2 21 11000 1000. 3 x xy y 3100 100

    Hay ni cch khc phn tch a thc nhn t ta c kt qu:

    3 2 2 2 2x 2x y xy y xy 3x 3y x y x xy y 3 Cch 2: S Hoorne:

    x 1 200 10103 10300

    100 1 100 103 0

    Vy 3 2

    2x 200x 10103x 10300 x 100x 103x 100

    Hay 3 2 2 2 2x 2x y xy y xy 3x 3y x y x xy y 3 . Ch : Phng php ny rt c ch cho cc bi ton v ch

    tng giao th hm s bc 3.

    IV. K nng 4: K nng tm max/min ca phn s

    Hng i 1: Tm max/min bng TABLE

    V d ta mun tm max/min ca 1

    x 2 2 :

    Vi chc nng TABLE ca my

    tnh Casio ta c:

  • TH THUT GII TON PHNG TRNH V T - ON TR DNG

    4

    1 1max 0.5

    2x 2 2

    Ch rng: maxA a th biu

    thc a A 0 lun ng.

    Do nu sau khi lin hp:

    Xut hin A , ta tm minA .

    Xut hin A , ta tm maxA .

    Hng i 2: S dng nh gi c lng:

    c lng theo s: c c

    b,c 0ba b

    .

    c lng theo bc cao nht: 2 2

    x 1 x 1

    2x 2x 5 x x x

    Ch : Ln hn hay nh hn chc chn ta s dng TABLE

    kim tra, iu ny gip khm ph ra nhng gi tr min/max kh c

    bit, chng hn nh sau:

    2 2

    2 2

    x x 2 x x x 1

    2x x 1 x x x

    Kim tra 2

    2

    x x 2 x 1

    2x x 1 x

    trong TABLE vi iu kin c c

    kim tra cn thn nhm biu thc ny dng hay m.

  • T TH THUT GII TON PHNG TRNH V T - ON TR DNG

    5

    CH 2: TNG QUAN CC PHNG PHP GII

    Cc phng php chnh khi gii ton phng trnh:

    1. T duy t n ph:

    t 1 n ph: Mc ch a v mt phng trnh, bt phng trnh

    c bn hn. Vy khi no t c n ph? Quan st h s, pht hin

    s lp i lp li:

    V d 5: 3

    22 25x 18x 9 5x 1 4 5 x 3 x 1

    32 2

    2 4x 9 x 1 4x x 1 4 5 4x 3 x 1 x 1

    Thng thng n bc ny cn phi quyt nh thc hin cc php

    bin i c bn a v n ph (Cng, tr, nhn, chia). Nu la chn

    php chia th phi trit tiu 1 bin: 2

    4x 4x 4 4x2 9 1 5 3

    x 1 x 1 x 1x 1

    Thng hc sinh hay nn nht bc quyt nh c n ph ha

    c hay khng ny, l cn bin i biu thc lc loi v c n

    ph cn t, v c th h s bt nh ha:

    4 16 4x

    x 1 x 1x 1

    Ti y ta quy ng v ng nht h s: 4 0 4

    16 16

    .

    Hay ni cch khc ta bin i phng trnh v dng: 2

    4x 4x 4x 4x2 9 1 16 4 5 3

    x 1 x 1 x 1 x 1

    n y bi ton c th x l c n gin hn rt nhiu. Mi bn

    c tip tc vi hai bi ton c bn p dng sau:

    p dng 1: 2

    2

    2

    3x 4x 8x 3x 6 x

    2 x 3x 6 x 4

    p dng 2:

    5

    23

    3 3

    3 x 2x x 2

    x 2x 4 x x 2

  • TH THUT GII TON PHNG TRNH V T - ON TR DNG

    6

    t 2 n ph tr ln: Mc ch nhm nhn t hoc s dng hm

    c trng. Bn cht ca hm c trng cng chnh l php t n

    ph, do nu ta t duy liu c hm c trng c hay khng, ta

    nn chuyn t duy thnh c th dn v hai n ph c hay khng?

    V d 6: 3 2 2

    2 2

    2x 3x 23x 11 3 x 4x 5x 1 0

    x 2x 2 x 4x 5

    Trc tin hc sinh cn bit rt gn phng trnh v dng:

    2 2 3 2x 1 x 2x 2 x 2 x 4x 5 2x 3x 23x 11 0

    Ti y, ta t duy xp hai cn sang hai pha v quan st d dng

    thy hai n ph:

    2 23 2x 1 x 1 1 2x 3x 23x 11 2 x 2 x 1

    Tuy nhin nh ti ni trn, kh khn nht lun l x l nhm

    biu thc cn li, v theo kinh nghim ca ti, l s dng phng

    php h s bt nh v ng nht h s: 3 22x 3x 23x 11

    3 3 2 2x 1 2 x x 1 2 x x 1 2 x tm cc h s, ngoi vic ph v biu thc v nhm theo tng bc

    ca bin x, ta c th thay 4 gi tr bt k ca x vo tm:

    x 1 27 9 3 391

    x 0 7 3 111

    x 3 65 15 5 851

    x 4 133 21 7 161

    Ti sao 3 n m cn 4 phng trnh? V cn c mt phng trnh

    kim tra ! Khng phi lc no cng ng u nh, nn phi ht

    sc cn thn !

    Vy ta vit li thnh: 2 3 2

    x 1 x 1 1 x 1 x 1 x 1

    2 3 2

    2 x 2 x 1 2 x 2 x 2 x

    p dng 3: 2 2x 1 x 2x 5 4x x 1 2 x 1 .

    (Trch Thi Th Trung Tm Diu Hin Cn Th 2016 Ln 1)

  • T TH THUT GII TON PHNG TRNH V T - ON TR DNG

    7

    2. T duy to hng ng thc:

    y l mt php bin i tay to bo nhng li gip ch rt nhiu.

    Nu xut hin: ab To ra 2

    a b .

    Nu xut hin: ab a b To ra 3

    a b .

    V d 7: 2

    32 2x x 1 2 x 1 x 2 1 x 1

    Phng trnh 2

    32 2x 2x 1 2 x 1 x 2 x 2 1 x 1

    22

    3 2x 1 x 2 1 x 1

    3 32 2x 1 x 2 1 x 1 x 1 x 1 x 2 1 0

    3 23 3x 1 x 2x 1 x 1 x 2 1 0

    3

    2 23 32 2 3 3

    x x 3 x 1 x 30

    x 2 1x 2x 1 x 2x 1 x 1 x 1

    x 3

    V d 8: 3 23 33x x 7 x x 7 7x 12x 5x 6

    3 23 33x x 7 x x 7 7x 12x 5x 6

    3 3 23 3x x 7 3x x 7 x x 7 8x 12x 6x 1

    3 33 3x x 7 2x 1 x x 7 2x 1 3 x 7 x 1

    3 3 2 2x 1 x 7 x 3x 2x 6 0 x 1 x 4x 6 0

    x 1.

    V d 9: 23 4 8x 9x

    2 2 x 1 1x 3x 2 2x 1

    iu kin xc nh: x 1. Bt phng trnh cho tng ng vi:

    22x 3 2 x 1 1 9x 4 2x 1x 3x 2 2x 1

  • TH THUT GII TON PHNG TRNH V T - ON TR DNG

    8

    2x 3 2 x 1 13x 2 2x 1

    x

    Do x 1. Do BPT 22x 3 2 x 1 1 3x 2x 2x 1

    2 2

    2 x 1 x 1 x 2x 1 2 x 1 x 1 0

    V: 2 2

    x 1 x 1 0; x 2x 1 0; 2 x 1 x 1 0 , x 1

    2 2

    2 x 1 x 1 x 2x 1 2 x 1 x 1 0

    Vy BPT xy ra th

    x 1 x 1

    VT 0 x 2x 1 x 1.

    x 1 0

    3. T duy i tm nhn t:

    A. Tm nhn t nghim n hu t c bn:

    Lin hp cn bc 2 Lin hp cn bc 3 Lin hp cn bc 3 2 2a b

    a ba b

    3 3

    2 2

    a ba b

    a ab b

    3 3

    2 2

    a ba b

    a ab b

    Ch : 22 2 2 21 1 1a ab b a b a b 0, a,b

    2 2 2 .

    Gi s phng trnh f x 0 c nghim x 3 v trong phng trnh

    c cha cn thc x 6 , khi vi x 3 x 6 3 .

    Vy nu s dng lin hp: x 6 9 x 3

    x 6 3x 6 3 x 6 3

    khi

    s xut hin nhn t x 3 v c th rt ra lm nhn t chung.

    Tuy nhin, v x 3 nn ta cng c th nh gi x 6 3 x .

    Vy nu s dng lin hp: 2 x 3 x 2x x 6

    x x 6x x 6 x x 6

    ta

    cng rt c nhn t x 3 .

    Nh vy bn cht ca phng php nhn lin hp l rt ra nhn t

    chung ch ra nghim ca phng trnh. Khi hai i lng a v b c

  • T TH THUT GII TON PHNG TRNH V T - ON TR DNG

    9

    gi tr bng nhau, ta c th s dng nhn lin hp gia hai i lng

    ny.

    Phng php nhn lin hp truy ngc du cp 1:

    Nu trong phng trnh hay bt phng trnh c cha a ng thi c nh gi a b th s dng lin hp:

    a a b a b a . V d: x 1 2 khi ta s dng lin hp:

    x 1 x 1 2 x 1 2 x 1 .

    Nu trong phng trnh hay bt phng trnh c cha 3 a ng thi 3 a b th s dng lin hp:

    23 3 3 3a b a b a a b a . V d: 3 x 5 2 khi ta s dng lin hp:

    3 3 3 3x 5 2 x 5 2 x 5 x 5 4 x 5 . Phng php nhn lin hp truy ngc du cp 2: Gi s bi

    ton cha x 3 v phng trnh c nghim x 1 . Khi ta nh

    gi nh sau: 2 2x 3 2 x 1 2x x 1 2x ...

    Do ta c th s dng cc phng n lin hp sau:

    2 x 1 x 2x x 2

    x 1 x 3x 1 x 3 x 1 x 3

    2 x 1 4x 34x x 3

    2x x 32x x 3 2x x 3

    3 24 22

    2 2

    x 1 x x 3x 2x 2x x 2x 1 x 3

    x 1 x 3 x 1 x 3

    3 242

    2 2

    x 1 4x 4x 4x 34x x 32x x 3

    2x x 3 2x x 3

  • TH THUT GII TON PHNG TRNH V T - ON TR DNG

    10

    Vic la chn lin hp no l mt ngh thut v ngi s dng

    lin hp trong qu trnh lm bi cn phi l mt ngh s, phi bit

    phi hp gia cc iu kin bi ton a ra ban u t quyt

    nh u l lin hp cn tm.

    V d 10: 3 x 2 3x 4 3 2x 1 x 3

    3 x 2 3x 4 3 2x 1 x 3

    2x 1 3 2