Bai viet cua tui yeu cau ban im lang doc that cham .ca khi da hieu… (***) Hi hi…bai nay jup cac ban lop 8 hieu them doi chut ve phuong trinh.cai tu phuong trinh thoat tien thi nghe hoi la mot chut nhung cac ban da tung lam ma khong biet.khi cac ban giai cac bai toan do lop 3,cac ban da su dung on so rui thiet lap he thuc sau do tim ra an so.o day he thuc do hay hai dang thuc duoc noi voi nhau boi dau “=” nguoi ta goi la “phuong trinh” va cac an so goi la bien “bien” va gia tri cua bien goi la nghiem cua phuong trinh.mot bai toan do co the rat nhieu an len ta co phuong mot an,hai an…o day ta chi xet phong trinh mot an “vi co n an thi phai co n phuong trinh lien he.tru phuong trinh nguyen” (***) Phuong trinh co ban nhat la phuong trinh bac mot.phuong trinh bac mot la phuong trinh co an cao nhat la mot tuc chua x.phuong trinh co dang :ax+b=0.de ton tai x trong thuong trinh thi a#o .cach jai phuong trinh nay nhu sau :chuyen b sang ve ben kia :ax=-b.vi a#o len ta chia ca hai ve cua phuong trinh cho a.ta duoc : x=-b/a.don jan dung khong cac ban.cu nghi moi thu deu don jan di hi hi…cac ban thu lam mot vai vi du coi: vd1.gpt: 15 – 8x = 9 – 5x ~ 15-9=8x-5x ~ 6=3x ~ x=2.vay pt co nghiem:x=2 vd2.gpt: 5 – (x – 6) = 4(3 – 2x) ~ 5-x+6=12-8x ~ 8x-x=12-6-5 ~ 7x=1 ~x=1/7 .vay pt co nghiem:x=1/7 Cac bai toan nay cac ban can luu y khi chuyen ve phai can than khong nham la ok.cac bai toan nay deu di tim he so a,b cua phuong trinh ax+b=0,rui cho x=-b/a.trong bai toan tren viec tim a,b that la don gian,nhung co mot so bai thi tim that kho.ta khong the tim theo cach thong dung,nhom tat ca cac thua so chua x de tim a,va cac thua so khong chua x de tim b duoc,ta thu coi vi du nay: vd3.gpt: 2004 x 2003 x1 1 2002 x2 − − =− − .giai theo cach thong thuong ta se lam nhu sau: x ( 1 2002 − 1 2003 − 1 2004 ) = 2 2002 −1− 1 2003 , den day ta dem wy dong no xe ra mot so rat to.neu bai toan khong cho cac so 2003,2003,2004 , ma cho cac so to gap nhieu lan thi sao ta khong the lam theo cach nay duoc.ta se lam nhu sau:ta di tim dai luong y nao do thoa man:y=ax+b va phuong trinh tro thanh :y.hangso=khong.suy ra y=0 “day goi la phuong trinh tich.trinh tich la phuong trinh gom tich cac da thuc nhan voi nhau bang 0 suy ra mot trong cac da thuc phai bang khong”.suy ra x=-b/a.do do ta lam bai nay nhu sau : 2− x 2002 +1= 1−x 2003 +1+1− x 2004 ~ 2−x+2002 2002 = 1−x+2003 2003 + 2004− x 2004 ~ ( 2004−x ) ( 1 2002 − 1 2003 − 1 2003 ) =0 ~ 2004−x=0 x=2004.vay ptco nghiem: x=2004 (***) nhu da noi o tren thi : y=-x+2004 va hang so bang 1 2002 − 1 2003 − 1 2003 no co gia tri bao nhieu ta khong wan tam.viec jai bai toan nay ta su dung mot so bien doi tuong duong chung ta da hoc,de rui chung ta duoc phuong trinh bac nhat:y=ax+b=0.nguoi ta goi cach giai nay la wy ve phuong trinh bac nhat va cung co rat nhieu phuong trinh can chung ta wy ve phuong trinh bac nhat.tui gioi thieu vai loai sau day “nhung phuong trinh nay dung cac cach bien doi cac ban da hoc de wy ve chi mot so loai co ban nhat thui”cac ban len thuoc vi toan hoc can kinh nghiem la chinh “chung ta dang hoc khong phai nghien cuu len khong can tim cac cach jai moi la,dai dong mat thoi gian”.cu the nhu sau: (***)Phuong trinh bac cao “lon hon 2”wy ve phuong trinh bac nhat: 1.phuong trinh bac 2:dang a x 2 +bx+c=0(a≠ 0) Phuong trinh nay co ba truong hop xay ra tuy vao ve so a,b,c : Truong hop 1:phuong trinh co nghiem “thuong la nghiem chan” cac ban phan tich se duoc 2 nhan tu “phuong trinh tich”rui ta suy ra duoc hai nghiem.cac ban se thac mac neu nghiem le thi sao “tui tra loi rang da la bai thi phai nghiem chan vi nghiem le cac ban chua hoc khong ai cho vuot wa truong trinh dung khong” tui lam vi du nhu sau : vd4gpt : x 2 +6x +5=0 x 2 +x +5x+5= 0 x ( x +1 ) +5 ( x +1 ) =0 ( x +1 ) ( x +5 ) =0 x +1=0 hoa c x +5=0 x=−1 hoa c x=−5.va y p t co ha ing hiem x1 =−1 ,x2 =−5 truong hop 2:phuong trinh dua duoc ve dang hang dang thuc hay phuong trinh co the tuong duong voi : a( x+ b 2a ) 2 =0 tuc truong hop nay : b 2 =4ac .thi phuong trinh co nghiem la : x=-b/2a .thu lam vi du nay la hieu ngay : vd5gpt : 2 x 2 + 4x+2=0 2 ( x 2 +2x+1 ) =0 2(x+1) 2 =0 x=−1.vay pt co nghiem x=−1 truong hop 3:phuong trinh nay khong co nghiem : thi ta phai chung minh no khac khong cu the no se lon hon 0 khi he so a lon hon khong va nho hon khong khi he so a nho hon khong.o day ta luu y mot dieu hien nhieu sau: a 2 ≥0 voi moi a “nho dau bang xay ra khi a=o “ ta se lam hai vi du cu the sau : Vd6gpt : x 2 + x+ 1=0vi : x 2 +2. 1 2 x + 1 4 + 3 4 =( x+ 1 2 ) 2 + 3 4 ≥ 3 4 > 0cholen phuongtrinh vonghie m Vd7gpt : −x 2 +x−1=0 vi :−x 2 +2 . 1 2 x− 1 4 − 3 4 =−( x− 1 2 ) 2 − 3 4 ≤− 3 4 <0 cholen phuong trinhvo nghiem. 2.phuong trinh bac 3,4,5…doi xung: Bac 3:dang: a ax 3 +b x 2 ±bx ± a=0(¿0) nhom dau voi cuoi va hai so giua voi nhau. a ( x 3 ±1 ) + b ( x 2 ± x ) = 0 ( x ±1 ) ( a x 2 ±ax+1 ) + ( x ± 1 ) bx=0 ( x ±1 ) ( a x 2 ± ax+bx+1 ) =0. den day ta duoc mot phuong trinh bac hai “giai tiep theo cach o tren” va mot phuong trinh bac nhat co nhiem ±1 . Bac 4:dang: a x 4 +b x 3 + c x 2 +bx+a=0 (a#0) . chia ca hai ve cho x 2 ta duoc : ax 2 +bx+c + b x + a x 2 =0 . nhom tuong tu nhu tren ta duoc : a ( x 2 + 1 x 2 ) +b ( x+ 1 x ) +c=0 a(x+ 1 x ) 2 +b ( x+ 1 x ) +c−2a=0 neu ta dat y= x+ 1 x =x−2. √ x . 1 √ x + 1 x +2= ( √ x− 1 √ x ) 2 +2≥ 2 . khi do pt tuong duong voi : ay 2 +by+c=0 day la phuong trinh bac 2 da jai o tren.nhung phai luu y chi lay nhiem y lon hon hoc bang 2 . tim ra duoc y rui ta tiep tui wy dong duoc phuong trinh bac hai rui giai phuong trinh bac hai.noi thi hoi dai rong nhung lam cung ngan thui.lam mot vi du nha: Vd8gpt: x 4 +3 x 3 − 8x 2 + 3x+1=0 ta thay khi x=0 thi phuong trinh vo nghiem cho len ta chia ca hai ve cho x 2 ta duoc : x 2 +3x −8+ 3 x + 1 x 2 = 0 x 2 +2 + 1 x 2 + 3 ( x+ 1 x ) −10 =0 (x+ 1 x ) 2 + 3 ( x+ 1 x ) − 10=0 y 2 +3y −10= 0 y 2 − 2y+ 5y−10= 0 y ( y−2 ) +5 ( y− 2 ) =0 ( y −2 )( y+ 5 ) =0 y =2 hoac y=−5 hai nghiem nay ta chi nhan duoc mot nghien : y=2 x+ 1 x =2 x 2 −2x+1=0 x=1 vay phuong trinh co nghiem : x=1. Bac 5,6 ta nhom dau voi cuoi …nhu cach nhom o tren rui dat cac he so ra ngoai tim cach jai tiep… 3.phuong trinh bac 3,4,5 khong doi xung : nham cac nghiem chan sau :-3, -2,-1,0,1,2,3 “ day la cac nghiem nguyen thuong bai toan cho vao”neu thay co nhiem thi phan tich theo nghiem do.cu the no duoc suy ra tu cach suy luon sau : neu ta dat ve trai la mot da thuc thi ve phai la so khong.neu da thuc phan tich duoc thanh cac nhan tu”co nhan tu la bac mot” khi nhan tu nay bang khong thi da thuc bang khong.nghiem cua da thuc bac mot chinh la nghiem ta vua nham o tren. Vd9gpt : x 3 – 12 = 13x ~ x 3 −13x−12=0 “bien doi ve dang ve trai mot da thuc ve phai la so khong”.ta thay khi x=-1 thi phuong trinh tuong duong voi 0=0.jo ta phan tich theo nghiem x=-1. ~ x 3 + x 2 −x 2 −x −12x−12= 0 x 2 ( x+ 1 ) − x ( x +1 ) −12 ( x+ 1 ) =0 ( x +1 ) ( x 2 − x−12 ) =0 .ta lai tay khi x=-3 thi phuong trinh x 2 −x−12=0 tuong duong voi 0=0.ta tiep tuc phan tich theo nghiem x=-3. ~ x ( x+1 ) ( ¿¿2+3x −4x−12)=0 ( x+1 ) { x ( x+3 ) −4 ( x+3 ) } =0 ( x+1 ) ( x+3 )( x− 4 ) =0 x+ 1=0hoac x+3= 0hoac x−4= 0 ¿ vay phuong trinh co ba nghiem :x1=-1,x2=-3,x3=4 (***) Phuong trinh chua an o mau. Ta thay bien “an” la dai luong suy bien cho len mau cua bieu thuc tao phuong trinh co the bang khong la cho phuong trinh khong ton tai “phuong trinh vo nghiem” , do do truoc khi giai phuong trinh ta phai dat dieu kien cho phuong trinh.cach jai dang bai tap nay cung dong gian : wy dong khu mau dua ve phuong trinh bac “da giai o tren” nhung luu y : wy dong can than tranh nhan va khi giai ja duoc nghiem,ket luon nghiem thi phai loai nghiem neu trung voi dieu kien. Ai khong hieu cho nao thi pm cho minh : sbatreo_ntka@yahoo.com hoac 0943393042 . phong trinh mot an “vi co n an thi phai co n phuong trinh lien he.tru phuong trinh nguyen” (***) Phuong trinh co ban nhat la phuong trinh bac mot .phuong trinh. man:y=ax+b va phuong trinh tro thanh :y.hangso=khong.suy ra y=0 “day goi la phuong trinh tich .trinh tich la phuong trinh gom tich cac da thuc nhan voi nhau