I Sophucz thoa man dieu kien tren va c6 modun nho nhat khi diem bieu dien cua no 6 tren (C) va gan goc O nhat, do la mgt trong hai giao diem cua O
Néu ÂO thi phuong trinh co 2 nghi^m phan bi#t B ^
Chuyen del: C A N B A C H A I C U A S O P H U ' C
P H L T O N G T R I N H B A C H A I Ạ Trpng tam kien thuc Ạ Trpng tam kien thuc
1. Dinh nghia: Cho sophuco), so phuc z thoa man = © ggi la 1 can bac hai cua (,)
2. T i m can bac hai:
a) Truong hgp co la so thuc thi co = a . a = 0 thi can bac 2 cua o la 0 . a = 0 thi can bac 2 cua o la 0
. a > 0: z^ = a< = > ( z + \Ja){z - Ja) = 0 . Co hai can bac 2 cua co la va va - V ^
. a < 0 : = a o (Z - iV^ ) ( Z + i V ^ ) = 0 : co hai can bac hai cua ^ la:
yf^i va
h. Truong h^p: o) = a + bi (b ^ 0)
Z = X + y i la can bac hai ciia co: (x + yi)^ = a + bi x ^ - v ^ = a x ^ - v ^ = a
<=> I ^ giai he ta co: Hai can bac hai cua co la hai so doi nhau 2xy = b 2xy = b
3. Phu(mg trinh bac hai:
Dang: AZ^ + B Z + Z = 0 voi A , B , C lasophuc ( A ^ O ) Cach giai: Lap: A = - 4AC Cach giai: Lap: A = - 4AC
Néu ÂO thi phuong trinh co 2 nghi^m phan bi#t - B - ^ - B - ^
' 2A ' 2A Voi 9 la mot can bac hai cua A
' 2A ' 2A Voi 9 la mot can bac hai cua A
CM y: Néu A la so thuc ducmg thi hai nghi^m la: Z, = — — — = — — -
Néu A la so thi^c am thi hai nghi^m la: Z , = — =
Néu A = 0 thi phuong trinh co nghi^m kep Z =
2A ' 2A
- B
2A B. Phuong phap giai cac dang co ban B. Phuong phap giai cac dang co ban
Phuang phap: Ap dung cac cong thuc lien quan de giai
Bai 1: Giai cac phuong trinh sau a) Z^ = -4 a) Z^ = -4
i7n
b) Z ^ + V 2 = 0
a
a) = - 4 = 4i" <=>(Z-2i)(Z + 2i) = 0 o Z = 2i hay Z = - 2 i
b) Z 2 + V 2 = 0 » Z ' = V 2 i^ o( Z - V 2 i ) ( Z + V2i) = 0<:>
Bai 2: Giai phuong trinh: Z^ = 5 - 12i D a t Z = x + yi ( x , y ) e R D a t Z = x + yi ( x , y ) e R Z^ = 5 - 1 2 i o x ^ - y ^ = 5 2xy = -12 G i i i x = 3,y = -2 x = - 3 , y = 2 z = ^ i Z = - ^ i
0 Phuong trinh co 2 nghi^m: Z , = 3 - 2i va Z j = -3 + 2i
Bai 3: Giai cac phuang trinh
a) Z^ + 4 Z + 5 = 0 b) 4 Z ^ - 2 Z + 1 = 0
Giai
a) A = 16 - 20 = - 4 = 4 i ^ Phuang trinh co hai nghỉm la = - i - ^ = -2 - i va
7 - 4 + 2i . . Z , = = -2 +1 Z , = = -2 +1
b) A = 4 - 1 6 = -12 = 121^ = (2^31 )^
Phuong trinh CO hai nghi^m la: Z, = = IzJ^ va Z^ = Bai 4: Giai cac phuong trinh Bai 4: Giai cac phuong trinh
a) ( l- i) Z 2- ( 5 - 3 i ) Z + 10 = 0 b) z 2+ 2 ( i- l) Z - 2 i = 0 • •
Giai
A = (5 - 3i)^ - 40(1 - i) = -24 + lOi = (1 + 5i)^ Phuong trinh co 2 nghif m: Phuong trinh co 2 nghif m:
5 - 3 i- ( l + 5i) 4 - 8 i - . . „ 5 - 3 i + l + 5i 6 + 2i , ^.
Z i = — — = — = 3 - 1 va Z , = = = 1 + 2i 2( l- i ) 2( l- i ) 2 2( l- i ) 2 ( 1- 1 ) 2( l- i ) 2( l- i ) 2 2( l- i ) 2 ( 1- 1 )
^) A = 4(i -1)^ + 8i = 0. Phuong trinh co nghỉm kep la: Z = 1 - i
^ki 5:
Giai phuang trinh: Z ^ - 2 c o s a . Z + l = 0
Chung minh rSng phuang trinh: Z* - 4 Z ^ + 14Z^ - 36Z + 45 = 0 co 2 nghiem la sóthuan ao la sóthuan ao