Thiết kế bài giảng đại số 10 nâng cao (tập 1) phần 2

ChirONq ill

PHlTdNG TRINH VA HE PHl/dNG TRINH

P h a n 1 . EOrolllirO VAIXr D E CUA C H I / O R T G
L

NOIDUNG
Ndi dung chfnh cua chuong gom:

Phuong trinh bae nha't : Giai va bien luan phuong trinh bae nhat cd chiia
tham so, phuong trinh quy vl phuong trinh bae nhit.
- Phuang trinh bae hai : Giai va bien luan phuong trinh bae hai, dinh If
Vi-et, mdt so phuong trinh quy vl phuang trinh bae hai.
He phuong trinh bae nhit va he phuong trinh bae hai.
II.

MUC Tl£U

1.

Kien thiirc

Hieu khai niem phuong trinh, phuong trinh tuong duong, phuang trinh
ha qua; bia't dugc cac phep bien doi tuong duong va phep bia'n doi cho phuong
trinh ha qua.
Nim viing cdng thiic va cac phuong phap giai phuong trinh bae nha't,
phuong trinh bae hai mdt in va he phuong trinh bae nhat, bae hai 2 in.
Hieu y nghia hinh hgc ciia cac nghiem ciia phuong trinh va he phuong
trinh bae nha't va bae hai.
2.

KT nang
Bia't each giai va bien luan :
+ Phuong trinh bae nha't va bae hai mdt in,
+ Phuang trinh dang |ax + b| = [ex + d| va phuong trinh chiia in d miu,
235


+ Phuang trinh trung phuang
+ He hai phuong trinh bae nha't 2 in (bing dinh thiic cip hai)
• Bia't each giai (khdng bien luan):
+ He ba phuang trinh bae nhit ha in,
+ He phuang trinh bae hai
Bia't giai mdt so bai toan vl tuang giao giiia dd thi cua hai ham sd bae
khdng qua 2.
3.

Thai do
- HS cd tfnh cin than, kidn tri va khoa hgc khi tim giao cua hai d6 thi.

HS tha'y dugc quan he mat thiet gifla toan hgc va ddi sdng, toan hoc
xua't hiendo nhu ciu tur ddi sdng.

236


P h a n a . CAC B A I SOAJV

§1. Dai cvtdng ve phtfofng trinh (tiet 1, 2)
L

MUC TifiU

1.

Kie'n thurc
Giiip HS :

Hiiu khai niem phuang trinh, tap xac dinh (diiu kien xac dinh) va tap
nghiem ciia phuong trinh.
- Hia'u khai niem phudng trinh tuong duong va cac phep bien doi tuong duong.
2.

KT nang

Biet each thii xem mdt so cho trudc cd phai la nghiem ciia phuang trinh da cho
hay khdng.
Bia't sii dung cac phep bia'n ddi tuong duong thudng diing.
3.

Thai do
Ren luyen tfnh nghiam tiic khoa hgc.

II.

CHUAN BI CUA GV VA HS

1.

Chuan bi ciia GV :

Chuin bi bai ki cac kia'n thiic ma HS da hgc d ldp 9 de dat cau hdi.
Chuin bi mdt sd hinh ve trong SGK; pha'n mau,...

2.

Chuan hi ciia HS :
Cin dn lai mdt so kie'n thiic vl ham sd da hgc d ldp 9.

ra.

PHAN P H 6 I T H 6 I

LUONG

Bai nay chia lam 2 tia't:
Tie't thut nhdt: tic ddu de'n he't phdn 3;
Tiet thvc hai: ta phdn 4 den het phdn bdi tap.

237


IV.

TIEN TRINH DAY HOC

A. Bai cii
c a u hdi 1
Hay cho bia't nghiem ciia phuong trinh 2x - 1 = x + 4
c a u hdi 2
Cac sd nao sau day la nghiem cua phuong trinh x^ + x = V2x,

1;2;-1;0

B. Bai mdi
HOATDONGI
1.

Khai niem phuang trinh mot an
Dinh nghTa. Cho hai ham sd y = f(x) va y = g(x) cd tap xac dinh lin lugt la

% va 2)g. Dat 3) = 3)f n 2)^.
Menh dl chiia bia'n f(x) = g(x) dugc ggi la mdt phuang trinh mdt in;
X ggi la dn 50'(hay dn) va 3) ggi la tap xdc dinh cua phuong trinh.
Sd XQ e S) dugc ggi la mdt nghiem ciia phuang trinh f(x) = g(x) ne'u
f(Xo) = g(Xo) la menh dl diing.
GV: Thuc hien thao tdc ndy tron^ 3 phut.
Hoat ddng cua GV

Hoat ddng ciia HS

c a u hdi 1
Cy'.fi y tra Idi cau hdi 1
Hay neu mdt vf du vl phuang Chang ban Vx +1 = x + 1 .
trinh mot in.
Ggi y tra Idi cau hdi 2
c a u hdi 2
Tap xac dinh ciia phuang trinh nay
Hay nau tap xac dinh cua
la [1; + 00).
phuang trinh vira nau

238


Ggi y tra Idi cau hdi 3
Cau hdi 3
Hay chi ra mdt nghiem ciia Chang ban x = 1 la nghiem.
phuang trinh.
Chu y 1
1) De thuan tien trong thuc hanh, ta khdng cin via't rd tap xac dinh ii) cua
phuong trinh ma chi cin neu diiu kien de x e 3). Diiu kien dd ggi la diiu kien
xac dinh ciia phuang trinh, ggi tit la diiu kien ciia phuang trinh.
Nhu vay, diiu kien ciia phuong trinh bao gom cac diiu kien de gia tri cu.a
f(x) va g(x) ciing dugc xac dinh va cac diiu kien khac ciia in (ne'u cd yau cau)
(theo quy udc vl tap xac dinh ciia ham sd cho bdi bieu thiic).
GV: Thuc hien thao tdc ndy trong 3 phdt.
Hoat ddng ciia HS

Hoat ddng ciia GV
c a u hdi 1

Ggi y tra Idi cau hdi 1

Hay ndu mdt thuat ngii khac Diiu kien xac^nh ciia phuong trinh
ve tap xac dinh cua phuang hoac diiu kien cua phuang trinh.
trinh.
c a u hdi 2

Ggi y tra Idi cau hdi 2
Hay neu md'i quan he giiia tap
nghiam va tap xac dinh cua Tap nghiam la tap con cua tap xac

dinh cua phuang trinh.
phuang trinh.

Vi du 1.
a) Diiu kien xac dinh ciia phuong trinh
V x ^ - 2 x ^ + 1 = 3 la x^ - 2x^ + 1 > 0.
b) Khi tim nghiem nguyan cua j)huong trinh 2

= Vx ta hieu diiu
X

kian ciia phuang trinh l a x e Z,X7tOvaj[:>0 (hay x nguyan duong).

239


Chu y 2.
1) Khi giai mdt phuang trinh (tiic la tim tap nghiem ciia phuang trinh), nhilu
khi ta chi cin, hoac chi cd the tfnh gia tri gin dung cua nghiam (vdi do chinh
xac nao dd). Gia tri dd ggi la nghiem gin dung cua phuang trinh.
Ching ban, bing may tfnh bd tiii, ta tfnh nghiem gin diing (chfnh xac da'n
hang phin nghin) ciia phuong trinh x = 7 la x « 1,913.
2) Nghiem ciia phuang trinh f(x) = g(x) chfnh la hoanh do cac giao diem
cua do thi hai ham sd y = f(x) va y = g(x).
GV: Chi neu rdt nhanh cdc nhdn xet ciia chd y tren.

HOATDONG2

2.

Phuong trinh tuong duang
Ta da bia't : hai phuang trinh tuang duang na'u chiing cd cung mot tap

nghiem. Neu phuang trinh fi(x) = gi(x) tuong duong vdi phuang trinh f2(x) =
g2(x) thi ta via't:
fi(x) = gi(x) ^

f2(x) = g2(x).

GV: Nhdn mqnh :
- Hai phuang trinh ma tap nghiam ciia phuang trinh nay bang tap nghiem
ciia phuong trinh kia thi tuong duong vdi nhau.
- Hai phuong trinh cung vd nghiem thi tuong duong.
GV: Hudng ddn HS thuc Men|H1| vd thuc Men thao tdc ndy trong 5 phdt.
Hoat ddng cua GV

Hoat ddng cua HS

c a u hdi 1
Ggi y tra Idi cau hdi 1
Tim tap nghiem cua phuang Tap nghiem ciia phuofng trinh nay la:
trinh :
S={1}
Vx-l=2Vl-x
240


c a u hdi 2

Ggi y tra Idi cau hdi 2

Tim tap nghiem cua phuang Tap nghiam ciia phuong trinh nay la
trinh :
{1}
X - 1 = 0.

c a u hdi 3
Ggi y tra Idi cau hdi 3
Hai phuang trinh nay cd Hai phuang trinh nay tuang duang,
tuang duang khdng? vi sao?
vi chiing cd ciing tap nghiem.
Cau hdi 4
Ggi y tra Idi cau hdi 4
Tim tap nghiem cua phuang Tap nghiem cua phuong trinh nay la 0
trinh
X + Vx-2 = 1 +Vx-2

Ggi y tra Idi cau hdi 5
c a u hdi 5
Tap nghiam cua phuang trinh nay la
Tim tap nghiem ciia phuang {1}
trinh
X - 1 = 0.
c a u hdi 6
Ggi y tra Idi cau hdi 6
Hai phuang trinh nay cd Hai phuang trinh nay khdng tuang
tuang duang khdng? vi sao?
duang, vi chiing khdng cd ciing tap
nghiem.
c a u hdi 7
Ggi y tra Idi cau hdi 7

Tim tap nghiem ciia phuang Tap nghiam ciia phuang trinh nay la
trinh
{-i;i}
1x1 = 1
Cau hdi 8
Ggi y tra Idi cau hdi 8
Tim tap nghiem ciia phuang Tap nghiem ciia phuong trinh nay la
trinh
{1}
x = 1.

le-TKBGBAISOIONC-TI

241


c a u hdi 9
Ggi y tra Idi cau hdi 9
Hai phuang trinh nay cd Hai phuang trinh nay khdng tuang
tuang duang khdng? vi sao?
duang, vi chiing khdng cd ciing tap
nghiem.
GV: Neu cdc nhdn xet sau :
Khi mudn nha'n manh hai phuong trinh cd ciing tap xac dinh ^ (hay co
ciing diiu kien xac dinh ma ta ciing kf hieu la 9^) va tuong duong vdi nhau,
ta ndi :
- Hai phuong trinh la tuong duong vdi nhau tren ID, hoac
- Vdi dieu kien if*, hai phuong trinh la tuong duong vdi nhau.
Chang ban: Ta ndi, vdi x > 0, hai phuong trinh x = 1 va x = 1 la tuong
duong vdi nhau.

Trong cac phep bia'n doi phuong trinh, dang chii y nhat la cac phep bien
doi khdng lam thay doi tap nghiem ciia phuang trinh. Ta ggi chiing la cac phep
bia'n doi tuong duang. Phep Men ddi tuang duang Men mot phuang trinh thdnh
phuang trinh tuang duang vdi nd.
Dudi day la dinh If vl cac phep bien doi tuong duong thudng diing.
Dinh Ii 1. Cho phuang trinh f(x) = g(x) cd tap xdc dinh iP; y = h(x) la
mot hdm sd xdc dinh tren iD (h(x) cd the Id mot hang sd). Khi dd tren y\
phuang trinh dd cho tuang duang vdi mdi phuang trinh sau :
\)f(x) + h(x) ^g(x) + h(x);
2)f(x) h(x) = g(x) h(x) ne'u h(x) ^ 0 vdi mgi A: e 2).
GV: Hudng ddn HS chicng minh nhanh dinh li.
Tit dinh If tran, ta dl thay : hai quy tic bien doi phuong trinh da hgc d ldp
dudi (quy tic chuyen ve va quy tic nhan vdi mdt sd khac 0) la nhiing phep bien
doi tuong duong.

242


GV: Hudng ddn HS thuc hien\H2\ vd thuc hien thao tdc ndy trong S phut.
Hoat ddng ciia HS

Hoat ddng cua GV
Cau hdi 1

Ggi y tra Idi cau hdi 1
O cau a), sau khi chuyen ve' Cd.
cd dugc phuang trinh tuang Theo dinh If tren.
duang hay khdng?
Ggi y tra Idi cau hdi 2
Sau khi luge bd ta dugc phuang

b) Cho phuang trinh
trinh 3x = x^ Phuang trinh nay cd
3x + V x - 2 = x^ + V x - 2 .
hai nghiam x = 0 va x = 3. nhung
Luge bd V x - 2 d ca hai ve X = 0 khdng phai la nghidm ciia
ciia phuang trinh thi dugc phuang trinh ban dau.
phuang trinh tuang duang.
Hai phuang trinh nay khdng tuang
duang.

cau hdi 2

HOAT DONG 3
Phuong trinh he qua
GV: Neu vi du 2 de ddt vdn de ve phuang trinh he qua.
Xet phuong trinh :
Vx = 2 - X.

(1)

Binh phuong hai ve, ta dugc phuong trinh mdi :
X = 4 - 4x + x^

(2)

Tap nghiem ciia (1) la Sj = {1}, cua (2) la S2 ={ 1; 4}. Hai phuong trinh
(1) va (2) khdng tuong duong. Tuy nhian ta thay S2 3 Sp Trong trudng hop nay
ta ggi (2) la phuong trinh he qua ciia phuong trinh (1).
Tong quat, fi(x) = gi(x) goi la phuang trinh he qua ciia phuong trinh
f(x) = g(x) neu tap nghiem cua nd chita tap nghiem cua phuong trinh f(x) = g(x).

Khi dd ta viet
f(x) = g(x)=^fi(x) = g,(x).
243


GV: Thuc hien thao tdc ndy trong 2 phiu.
Hoat ddng cua GV

Hoat ddng cua HS

c a u hdi 1
Ggi y tra Idi cau hdi 1
Hay ndu vf du ve hai phuang x = l = > x ^ = l .
trinh he qua:

1

c a u hdi 2
Ggi y tra Idi cau hdi 2
Hay chi ra nghiem ngoai lai.
Nghiem ngoai lai cua phugng trinh
(nghiem ngoai lai ciia phuang la-1."
trinh la nhiing nghiem ciia
phuang trinh he qua ma
khdng phai la nghiem cua
phuang trinh ban diu)
Tii dinh nghia nay, ta suy ra : Neu hai phuang trinh tuang duang thi mdi
phuang trinh deu la he qua ciia phuang trinh con lai.
Trong vf du 2, gia tri X = 4 la nghiem cua (2) nhung khdng la nghiem ciia (1).
Nan X = 4 la nghiem ngoai lai cua phuang trinh (1).

GV: Huang ddn HS thuc hien |H3| vd thuc hien thao tdc ndy trong 2 phiit.
Hoat ddng ciia GV

Hoat ddng ciia HS

c a u hdi 1
Ggi y tra Idi cau hdi 1
Khing dinh sau day diing hay Diing vi hai phuong trinh nay tuong
sai?
duong.
a) V x - 2 = 1 = > x - 2 = 1?
Ggi y tra Idi cau hdi 2
c a u hdi 2
Khang dinh sau day diing Diing, vi phuang trinh diu vd
nghiem con phuang trinh sau cd
hay sai?
nghiem x = 1.
, X x(x-l)

b) -5^
^ = l = > x = 1?
x-1
Trong cac phep bie'n doi chi cho phuang trinh he qua, dang chu y la phep
bia'n doi dugc nau trong dinh If sau.
244


Djnh II 2. Khi binh phuong hai ve cua mdt phuong trinh, ta dugc phuong
trinh he qua cua phuong trinh da cho. Ndi each khac :
f(x) = g ( x ) ^ [ f ( x ) f = [g(x)f

GV: Thuc hien thao tdc ndy trong 5 phut.
Hoat ddng cua HS

Hoat ddng cua GV

Ggi y tra Idi cau hdi 1
c a u hdi 1
Hay ndu mot vi du. ap dung Vx +1 = 2x +1 ^ X +1 = f 2x +1 j ^
dinh If 2.
c a u hdi 2
Ggi y tra Idi cau hdi 2
Hay chiing to day la phep
Phuang trinh dau chi cd nghiem
bia'n doi he qua.
X = 0; nhung phuang trinh sau cd
3
hai nghiam x = Ova x =
4
Chii y
1) Cd the chiing minh ring na'u hai va' ciia mdt phuang trinh ludn cung
ddu thi khi binh phuong hai ve' cua nd, ta dugc phuong trinh tuong duong.
2) Na'u phep bien doi mdt phuong trinh din den phuong trinh he qua thi
sau khi giai phuong trinh he qua, ta phai thit Iqi eae nghiem tim dugc vao
phuong trinh da cho de phat hien va loai bd nghiem ngoai lai.
GV: Thuc Men thao tdc ndy trong 2 phiit.
Hoat ddng ciia GV

Hoat ddng ciia HS

Ggi y tra Idi cau hdi 1

c a u hdi 1
Hay dat diiu kien cho x de khi
binh phuong hai va' ciia phuong Diiu kien x > —
2
trinh Vx - 1 = 2x +1 ta dugc
phuong trinh tuong duong.
c a u hdi 2
Tim nghiem eiia phuong trinh.

Ggi y tra Idi cau hdi 2
Binh phuang hai ve' ta dugc :
245


3
4x^ + 3x = 0 o X = 0 va x = - - Kiem
4
tra dieu kien ta duoc x = 0 la nghiem.
GV: Neu vd hitdng ddn HS gidi vi du 3 trong SGK hoac cd the thay baitg vi du
khdc tuang tie
HOATDONG 4
4.

Phuong trinh nhieu an

Trong thuc te, ta con gap cac phuong trinh cd nhilu ban mot in. Dd la
cac phuong trinh dang F = G, trong dd F va G la cac bieu thiic cua nhilu bie'n.
Vi du :
2x + 4xy - y^ = -X + 2y + 3

(3)

la mdt phuang trinh hai dn (x va y);
X + y + z = 3xyz

(4)

la mdt phuang trinh ba an (x, y va z).
Neu phuong trinh hai in x va y trd thanh menh dl diing khi x = Xy va y = y,, (vdi
Xo va y„ la cac sd) thi ta ggi cap sd (x^{, y,,) la mdt nghiem ciia phuong trinh.
Ching ban, cap sd (1; 0) la mdt nghiem ciia phuong trinh (3).
Khai niem nghiem ciia phuong trinh ba in, bd'n an,... ciing dugc hieu
tuong tu. Ching han bd ba sd (1; 1; 1) la mdt nghiem cua phuong trinh (4). Ddi
vdi phuong trinh nhieu in, cac khai niem : tap xac dinh (diiu kien xac dinh),
tap nghiem, phuong trinh tuong duong, phuong trinh he qua,... ciing tuong tu
nhu ddi vdi phuong trinh mdt in.
GV: Thitc hien thao tdc ndy trong 5'
Hoat ddng cua GV

Hoat ddng ciia HS

c a u hdi 1
Ggi y tra Idi cau hdi 1
Hay neu mot vi du ve phuang X + y = 5.
trinh 2 an.
c a u hdi 2
Hay chi ra mdt nghiem ciia nd.
246

Ggi y tra Idi cau hdi 2

(0;5),(1;4),...


c a u hdi 3
Ggi y tra Idi cau hdi 3
Hay ndu mdt vf du ve phuang X + y = xy.
trinh 3 in.
c a u hdi 4
Ggi y tra Idi cau hdi 4
Hay chi ra mot nghiem cua (0; 1),(0;2),(1;0),...
phuang trinh.
HOATDONG 5
5.

Phuong trinh chiia tham sd

Chung ta con xet ca nhiing phuong trinh, trong do ngoai in x con cd
nhiing chii khac. Cac chii nay dugc xem la nhiing sd cho trudc va dugc ggi la
tham sd.
Ching ban, phuong trinh m(x + 2) = 3mx - 1 la mdt phuong trinh chiia
tham sd m.
GV: Hudng ddn HS thuc hien\\\'5\ vd thuc Men thao tdc ndy trong 5 phiit.
Hoat ddng cua GV

Hoat ddng ciia HS

Ggi y tra Idi cau hdi 1
cau hdi 1
Khi m = 0, hay chi ra phuang 2 = 1 ; Tap nghiem la 0 .
trinh va tap nghiem.

cau hdi 2
Ggi y tra Idi cau hdi 2
Hay chi ra mdt nghiem
phuang trinh khi m ^ 0.
m

r"-'i.

Rd rang nghiem va tap nghiem ciia mdt phuang trinh chiia tham so phu
thudc vao tham so. Khi giai phuang trinh chiia tham so, ta phai chi ra tap
nghiem ciia phuang trinh trong mdi trudng hgp cd the cua tham sd. De nhain
manh y dd khi giai phuang trinh chiia tham so, ta thudng ndi la gidi vd Men
ludn phuang trinh.

247


TOM TAT BAI HOC
1.

Cho hai ham sd y = f(x) va y = g(x) cd tap xac dinh lin lugt la iDf va 9")^.
Daty) = ii)f n

^\.

Menh dl chiia bien f(x) = g(x) dugc ggi la mdt phuong trinh mdt in;
X ggi la in sd (hay in) va 9) ggi la tap xac dinh ciia phuang trinh.
Sd XQ e ^ ggi la mdt nghiem ciia phuong trinh f(x) = g(x) na'u f(Xo) =
g(Xo) la menh di dung.
2.

Hai phuong trinh tuong duong ne'u chiing cd cung mdt tap nghiem. Neu
phuong trinh fi(x) = gi(x) tuong duong vdi phuang trinh f2(x) = g2(x) thi
ta via't:
fi(x) = gi(x)c^ f2(x) = g2(x).

3.

Dinh li 1.
Cho phuong trinh f(x) = g(x) cd tap xac dinh iT); y = h(x) la mdt ham sd xac
dinh trdn 9) (h(x) cd the la mdt hing so). Khi do trdn ^, phuong trinh da cho
tuong duang vdi mdi phuang trinh sau :
l)f(x) + h(x) =g(x) + h(x);
2) f(x) h(x) = g(x) h(x) neu h(x) 9^ 0 vdi mgi x e ^J).

4.

fj(x) = gi(x) ggi la phuong trinh he qua ciia phuang trinh f(x) = g(x) neu
tap nghiem ciia nd chiia tap nghiem ciia phuong trinh f(x) = g(x). Khi do
ta viet
f(x) = g(x)=>fi(x) = gi(x).

5.

Dinhli2.
Khi binh phuong hai vi ciia mdt phuong trinh, ta dugc phuong trinh he
qua cua phuong trinh da cho. Ndi each khac :
f(x) = g ( x ) ^ [ f ( x ) f = [g(x)]^

248

HUdNG

DAN T R A LCJI C A U

HOI vA

BAI T A P

SGK

Bail.
GV: Hudng ddn cdu a)
Hoat ddng cua HS

Hoat ddng ciia GV

Ggi y tra Idi cau hdi 1
c a u hdi 1
Tim dieu kien xac dinh ciia fx^O
<^
=:>X = 0
phuang trinh.
[-x>0
c a u hdi 2
Ggi f tra Idi cau hdi 2
Tim tap nghiem ciia phuang
{0}.
trinh.

Trd Idi:
Dieu kien ciia phuang trinh

Tap nghiem

a) Vx = V-x

x=0

{0}

b) 3 x - V x - 2 = V 2 - x + 6

x=2

{2}

x>3, x<3vax;t3
Khdng cd x nao

0

X > 1 va X < 0,

0

V3-X
c)

1 r
=x+ vx-3

x-3
d) X + Vx - 1 = V-x

Khdng cd x nao

Bai 2.
GV: Hudng ddn cdu a)
Hoat ddng ciia GV

Hoat ddng ciia HS

Ggi y tra Idi cau hdi 1
c a u hdi 1
Tim diiu kien xac dinh cua X > 1 .
phuang trinh.
Ggi y tra Idi cau hdi 2
c a u hoi 2
Tim tap nghiem cua phuang Phuang trinh vd nghiem.
Tap nghiem 0 .
trinh.

249


Trd Idi:
b) Vdi dieu kien x > 1 ta cd :
x + V x - 1 =0,5 + Vx-1

<=> X = 0,5 (loai vi khdng thoa man diiu kien x

> !)•

Vay phuong trinh vd nghiem.
c) X = 6;
d) Vd nghiem.

Bai 3.
GV: Hudng ddn cdu a)
Hoat ddng ciia GV

Hoat ddng ciia HS

Cau hdi 1
Ggi y tra Idi cau hdi 1
Tim diiu kian xac dinh ciia x # l .
phuang trinh.
c a u hdi 2
Ggi y tra Idi cau hdi 2
Tim tap nghiem ciia phuang
Vdi diiu kien x ;^ 1, ta cd :
trinh
1
2x-l
x+
=
x-1
x-1
<^

<:^

jc^-jc+l=2x-l

'x = l
_x = 2

Dd'i chieu vdi diiu kien ta ket luan:
phuang trinh cd nghiem x =2.
Trd Idi:
lb) Vdi dieu kien x ^t 2, ta cd :

250


x+- i _ ^ 2 x _ 3 ^
x-2
x-2

x^-2x+l=2x-3

c^ X = 2 (loai do diiu kien x T^ 2)
Vay phuong trinh vd nghiem.
c) Dieu kien : x > 3.
Da tha'y X = 3 la mdt nghiem.
Neu X > 3 thi X - 3 ^ 0. Do dd :
(x^ - 3x + 2) V x ^ = 0 o

x^-3x + 2 = 0

<» X = 1 hoac X = 2 (hai gia tri diu bi loai do diiu kian x > 3).
Vay phuong trinh cd mdt nghiem x = 3
d)xe{-l;2}.
Chu y : Binh phuong hai v6 ta chi thu dugc phuong trinh he qua nan
phai thii lai de ket luan nghiem.

Bai 4.
GV : Hudng ddn cdu a)
Hoat ddng ciia GV

Hoat ddng cua HS

cau hdi 1

Ggi y tra Idi cau hdi 1
Tim dieu kien xac dinh ciia 3 < A' < 4,5
phuang trinh.

Ggi y tra Idi cau hdi 2
c a u hdi 2
V x - 3 = V9-2x = > x - 3 = 9
Tim tap nghiem cua phuang
- 2x ^ X = 4.
trinh.
Thii lai thay diing. Vay phuong
trinh cd nghiem jc = 4.
Trd Idi:
b) Ta cd :

251


Vx-l = x - 3
=» X - 1 = (x - 3)^ => x^ - 7x + 10 = 0
=> X = 2 hoac X = 5.
Thii lai, gia tri x = 2 khdng thoa man.
Vay phuong trinh cd nghiem x = 5.
c) Ta cd :
2| X - 1 1 = X + 2
^ 4(x -1)^ = (X + 2f ^ 3x^ -12x = 0
=> X = 0 hoac X = 4.
Thii lai thay ca hai diu nghiem diing.
Vay phuang trinh cd hai nghiem x = 0 va x = 4.
d) Ta cd :
1 X - 2 1 = 2x - 1
^ (x - 2)^ = (2x - 1)^ =^ 3x^ = 3
=>x = ± 1.
Thii lai, ta thay chi cd x = 1 nghiem ddng.
Vay phuong trinh cd nghiem x = 1.

252


MOT SO B A I T A P T R A C NGHIEM
1.

Cho phuang trinh : x^2 + 1- =

1

Vx^'

Diiu kien ciia phuong trinh la
(a)R;

(b)xGR, x > l ;

(c)xeR, x>l;

(d)xeR, x ^ l .

Hay chgn ket qua diing.
2.

Phuong trinh |x| + 1 = x +Vx .
Trong cac sd sau day so nao la nghiem cua phuong trinh

3.

(a)-2;

(b)-l;

(c)l;

(d)0.

Trong cac phuang trinh sau, phuong trinh nao tuong duong vdi phuong trinh
x^ = 1

4.

(1)

(a) x^ + 3x - 4 = 0;

(b) x^ - 3x - 4 = 0;

(c)|x|=l;

( d ) x ^ + V x = 1+Vx

Cho phuong trinh
x^ + x + V x + l = 0 .

(1)

Hay diin diing - sai vao cac ket qua sau day:
(a)(l)<::>x^ + x + l = - V x

DDiing

QSai

(b)(l)c^x^ + x + 1 + V x ^ =-Vx + V x ^

DDiing

DSai

(c)(l)<::>x+1 + — + - = 0

Doling

DSai

DDiing

DSai

X

(d)(1) <:^x^ = - l
5.

X

Cho phuong trinh
Vx + 1 = x . ^

(1)
253


Hay chgn diing - sai trong cac khang dinh sau
( a ) ( l ) o x + 1 =x^

DDung

DSai

(b)(1) < ^ x + 1 =x^

DDung

DSai

(c) (1) <^ Vx + 1 + Vx = x^ + Vx

DDiing

DSai

(d)(1) <=> VVxTT =Vx

DDiing
DSai.

Cho phuong trinh
x^ + ( m - l)x + m - 2 = 0

(1)

Hay chgn ket luan diing trong cac ket luan sau
(a) phuong trinh (1) vd nghiem Vm;
(b) phuong trinh (1) cd 3 nghiem Vm;
(c) phuong trinh (1) cd 2 nghiem la x = - 1 va x = 2 - m;
(d) ca ba ket luan tren diu sai.
Cho phuong trinh

X+

1
=1
x+ 1

(1)

(a) phuong trinh (1) cd 2 nghiem la x = - 1 va x = 0;
(b) phuong trinh (1) cd 2 nghiem la x = 2 va x = 0;
(c) phuong trinh (1) cd 2 nghiem la x = 1 va x = 2;
(d) ca ba ket luan tran diu sai.
Hay chgn khing dinh diing.

Ddp dn:

254

l-(b)

2. (b)

3. (c)

4.(a) D,

(b)D,

(c)D,

(d)D

5. (a) Sai;

(b) Diing;

(c) Sai;

(d) Diing

6. (e).

7. (d).


BAITAPTUGIAI
8.

Cho phuong trinh
x^ + 2005x + VxTI = 1

(1)

Hay chgn ket luan dung trong cac ket luan sau.
(a) X = 1 va X = 5 la nghiem cua (1);
(b) X = 1 la nghiem ciia (1);
(c) X = 1 va X = - 1 la nghiem ciia (1);
(d) X = 1 va X = -1782 la nghiem ciia (1).
9.

Cho phuong trinh
x+.l=-^
x+1

(1)

( a ) ( l ) « x ^ + 3x + 2 = 0;
(b)(l)o(x-l)(x-2);
( c ) ( l ) o ( x - l ) ( x + 2);
(d)(l)o(x+l)(x-2).
Hay chgn ket qua diing.
10.

Cho phuong trinh
'

x + Vx = 0

(1)

( a ) ( l ) o x = O v a x = 1;
( b ) ( l ) c ^ x = 0;
(c)(1) < » x = 1 vax = 2;
(d)(l)c^x = 0vax = -2.
Hay chgn ket qua diing.
11.

Cho phuong trinh

v;^

2 . /- .
X + vx +

1

1

x-3

Vx-2
255


(a) Diiu kien ciia phuong trinh la : x > 0, x 9^ 3;
(b) Diiu kien ciia phuong trinh la : x > 2;
(c) Diiu kien eiia phuong trinh la : x > 2, x ?!: 3;
(d) Diiu kien ciia phuong trinh la : x > 0.
Hay chgn ket qua diing.
12.

Cho hai phuong trinh : |x|= 1 (1) va x^ - 3x + 2 = 0 (2)
(a) (1) la ha qua ciia (2);
(b) (2) la he qua cua 1);
(c)(l)«(2);
(d) ca ba ket luan tren diu sai.
Hay chgn ket qua diing.

13.

Cho hai phuong trinh
x+l + ^ i = = - 2
Vx + 1

(1)

va
x^ + 2x + 5 = 0
Hay chgn ket qua diing trong cac khing dinh dudi day:
(a) (1) la he qua cua (2);
(b) (2) la he qua cua 1);
(c)(l)c^(2);
(d) ca ba ket luan tran diu sai.
14.

256

x2
1
.
= .
cd nghiem la
Vx-1
Vx-1
(a) X = 1 va X = - 1 ;
(b) X = 1 va X = 0;
(c)x=l;
(d) X = 1 va X = - 2 .
Hay chgn ket qua diing.

Phuong trinh

(2).


15.

Phuang trinh

^

Vx^

Vx^"

(a) cd nghiem x = 2;
(b) vd nghiem;
(c) cd nghiem x = 3;
(d) ca ba ka't luan tran diu sai.
Hay chgn ket qua diing.
16.

Cho phuang trinh : x"^ + 3x^ + 2 = 0.
(a) Phuang trinh cd 2 nghiem x = - 1 va x = - 2 ;
(b) Phuong trinh cd 4 nghiem x = ±1 va x = ±2;
(c) Phuang trinh vd nghiem;
(d) ca ba ket luan tran diu sai.
Hay chgn ka't qua diing.

17.TKBGOAIS610NC-T1

§2. Phufdng trinh bae nhat va bae hai mot an
•

'V^i

(tiet 3, 4)
L

MUC TIEU

1.

Kien thirc
Cling cd kia'n thiic vl va'n dl bien doi tuong duong cac phuong trinh.
Nim dugc cac iing dung ciia dinh If Vi-et.

2.

KT nang
Nim viing each giai va bien luan phuong trinh bae nha't: ar + b = 0 va bae hai

ax + bx + c = 0.
Bia't each bien luan sd giao diem cua mdt dudng thing va mdt parabol
va kiem nghiem lai bing do thi.
Biet ap dung dinh If Vi-et de xet dau ciia mdt phuong trinh bae hai va
bien luan sd nghiem cua mdt phuong trinh triing phuang.
3.

Thai do
• Ren luyen tfnh cin than, dc tu duy Idgic va to hgp.

IL

CHUAN BI CUA GV VA HS

1.

Chuan bi cua GV :
Chuin bi bai kT cac kien thiie ma HS da hgc d ldp 9 de dat cau hdi.
Chuin bi mdt sd hinh ve trong SGK va pha'n mau,...

2.

Chuan bj ciia HS :
• Can dn lai mdt so kien thiic vl ham sd da hgc.
Can dn lai phin phuong trinh da hgc d ldp 9 va bai 1.

258


in.

PHAN

PHOI THCII LI/ONG

Bai nay 2 tiet :
Tie't thic nhdt: tu: ddu de'n he't phdn 3;

Tie't thic hai: td phdn 4 de'n he't phdn bdi tap.
IV. TIEN TRINH DAY HOC
A. Bai cii
cau hoi 1
Hay tim nghiem cua cac phuong trinh sau :
a) X - 1 = Vx - 1 ;
b ) x ' - 3 x + 2=0;
c) V 3 - X = 4 x-3
cau hoi 2
Phuong trinh mx^ + x - 1 = 0 ludn cd hai nghiem. Diing hay sai?
B. Bai mdi
HOATDONGI
GV: Ddt van de :
Phuang trinh bdc nhdt (mdt an), tiic la phuong trinh cd dang
ax+ b = 0
(a va b la hai sd da cho vdi a ^ 0);
Phuang trinh bae hai (mot in) la phuong trinh ed dang
ax + bx + c = 0
(a, bvacla

ba sd da cho vdi a ^ 0);
2

Ta cd: A = b^
b - 4ac ggi la Me
Met thitc. A' = b'^ - ac (vdi b = 2b') ggi la Met
thdc thu ggn eiia phuong trinh bae hai.

259