PHNG PHP DNG LNG LIN HP GII PHNG TRNH V T

PHNG PHP DNG LNG LIN HP GII PHNG TRNH V T

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I. Mt s kin thc cn nh:

I.1. Mt s hng ng thc hay s dng:

+

+

+

+

S dng nhng hng ng thc ny, ta c th quy phng trnh v t ban u v dng phng trnh tch bng vic lm xut hin cc nhn t chung. T ta c th d dng gii quyt tip!

Thng th cc bi ton s dng phng php ny th tng tng qut ca ta nh sau: Gi s nu ta c phng trnh dng vi xc nh trn mt min D no v ta nhm c mt nghim x = a ca phng trnh th ta c th bin i phng trnh cho li thnh . n y ta ch vic x l phng trnh G(x) = 0 na l n! (Vic x l phng trnh G(x)= 0 c th s dng cng c o hm hoc bng bt ng thc).II. Cc v d minh ha:

Sau y, lm r thm ni dng v tng ca phng php, mi cc bn cng th sc vi cc v d sau:

II.1. Cc bi ton m uCc bn hy th sc mnh vi cc bi ton ny trc nh! Bi ton 1: Gii phng trnh sau:

Bi ton 2: Gii cc phng trnh sau:

a)

b)

c)

d)

II. 2. Bi tp minh ha:V d 1: Gii phng trnh

EMBED Equation.DSMT4 (1)

Gii:Ta d on c nghim , v ta vit li phng trnh nh sau:

Mt khc, ta c:

Nn phng trnh thc hai v nghim.

Vy (1) c 2 nghim .

V d 2: Gii phng trnh sau

(2)Gii: tng: Trc ht, kim tra ta thy c rng phng trnh cho c mt nghim nn ta s c gng a phng trnh trn v phng trnh tch xut hin nhn t . Ta c nhn xt rng:

v

Ta i n li gii nh sau:

(2)

Mt khc, ta c:

> 0 vi mi xVy phng trnh (2) c mt nghim duy nht x = 2.

V d 3: Gii phng trnh

EMBED Equation.DSMT4 (3)

Gii:Cng bng cch kim tra, ta thy pt (3) nhn x = 1 lm mt nghim nn ta c th a phng trnh (3) v dng phng trnh tch xut hin nhn t .Ta vit li nh sau:

(4)

rng hai phng trnh v v nghim nn nhn lin hp hai v ca (4) ta c:

Pt (*)

n y ta c hai hng gii quyt:

Hng 1: bnh phng hai v(Hng 2: kt hp vi pt (3) ta c h sau

Ly phng trnh th nht tr i 9 ln phng trnh th hai, ta thu c:

Vy phng trnh cho c 2 nghim .

V d 4: Gii phng trnh

Gii:Phng trnh cho tng ng vi:

EMBED Equation.DSMT4

Xt phng trnh:

Ta t suy ra:

EMBED Equation.DSMT4 Vy phng trnh cho c mt nghim duy nht .

V d 5: (Olympic 30/4 ngh)

Gii phng trnh sau:

Gii:k:

Ta nhn thy x = 2 l mt nghim ca phng trnh. Nh vy phng trnh cho c th phn tch c v dng !

Phng trnh cho tng ng vi:

Do nn pt (*) v nghim.Vy phng trnh cho c nghim duy nht x = 2.

V d 6: Gii phng trnh

Gii:Phng trnh cho tng ng vi

Bng cch nhn lin hp, ta c:

.

Do nn phng trnh c nghim duy nht x = 2.

V d 7: Gii phng trnh

EMBED Equation.DSMT4 Gii:K: .Phng trnh cho tng ng vi:

EMBED Equation.DSMT4

Vy phng trnh cho c mt nghim duy nht x = 1.

V d 8: Gii phng trnh

Gii:k:

EMBED Equation.DSMT4 ,

bi ny, kh l ch ta khng th nhm ra ngay c nghim ca phng trnh dng lng lin hp. Tuy nhin vi s h tr c lc ca cng ngh l chic my tnh Casio fx570 Es th mi chuyn c v d dng hn!Tht vy, ta s ln lt dng chc nng Shift Solve tm ra 2 nghim ca phng trnh l: sau gn hai nghim ny vo hai bin A v B.

By gi ta s th tm xem A v B c mi quan h g vi nhau hay khng bng cch tnh A + B v AB, ta thu c kt qu p sau: .

iu chng t A, B l hai nghim ca phng trnh:

V t y, ta c th d on c chnh l nhn t ca pt! (Ta vit pt cho li thnh:

n y, xut hin nhn t th vi l mt h s. Chn = 4 th ta c mt cp (p, q) tha mn l (p, q) = (-1; 2). Khi (2) tr thnh:

Xt ta c:

Ta c bng bin thin:

kt hp vi

Vy phng trnh cho c nghim .

V d 9: Gii phng trnh

EMBED Equation.DSMT4 Gii:Cng bng cch lm nh V d 8 (, ta phn tch c nh sau:

.

Ta cng c th gii thch theo cch khc ti sao li tm c lng nh sau:Do x = -2 khng l nghim ca phng trnh nn chia hai v phng trnh cho (x + 2) ta c:

. Gi s ta cn thm vo hai v ca phng trnh mt lng , khi ta c:

Khi , ta cn chn A, B sao cho . T ta c: A = 0, B = 3.V d 10: Gii phng trnh

Gii:K: .

Phng trnh cho tng ng vi:

.

- Vi bi ny, vic xut hin thm cc a thc cha tr tuyt i tng chng nh s gy cho ta thm kh khn trong vic gii quyt. Nhng nh s dng phng php nhn lng lin hp, bi ton c gii quyt nhanh chng! Khi y, ta ch cn chuyn cc lng trn v ng v tr v s dng pp nhn lin hp l .Sau y l mt s bi ton khc:

V d 11: Gii phng trnh

EMBED Equation.DSMT4 Gii:k:

Phng trnh cho tng ng vi:

.

III. Bi tpGii cc phng trnh sau:

(1)

S:

Hng dn: , trc cn thc lm xut hin nhn t chung x 1.

(2) S:

Hng dn: , sau trc cn thc lm xut hin nhn t chung x 3.

(3)

S:

Hng dn: , trc cn thc lm xut hin nhn t chung l .

(4)

S:

Hng dn:

(5)

S:

Hng dn: , trc cn thc lm xut hin nhn t chung l x 3.

(6)

S:

Hng dn: sau trc cn thc lm xut hin nhn t chung x 2.

(7)

S:

Hng dn: , trc cn thc lm xut hin nhn t chung l .

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