tai lieu on thi dai hoc mon toan sang tao va giai phuong trinh bat phuong trinh he phuong trinh bat dang thuc nxb dai hoc quoc gia 2014 2 4085

^ ^'^•^ dieu ki?n P = x.y b) Dat S ( s - P ) = 19 > 4P h§ phucmg trinh da cho tro thanh: SP = -8S = -8S =l S ^ - ( - S ) = ^ | S + S - = O ' ^ P = -6 SP S(8 + P) = S V i dv 2: G i i i cac h$ phuong trinh sau: a) b) D^t fS = a + b P = ab h? da cho tro tharJi: dieu kif n 1+ d) xV; = ( a ^ + b ) = 3(a2b + b2a) a+b=6 2xy x+y x^y + x^l + y + y^ j + y - 1 = Giii: a) Dgt Vx = a,7y = b dieu ki?n a,b > > 4P thi h^ da cho tro [ ( - P ) = 3P phuong trinh tro thanh: S=6 [S = trinh thanh: P =8 S=6 Suy a,b la nghi^m cua phuong trinh: x - X + = o X i = 2;X2=4=> a = 2=>x = fa = 4=>x = 64 b = 4=i>y = 64 [b = = > y = Vay h f da cho c6 hai cap nghi?m (x;y) = (8;64),(64;8) xy >0 x,y>-l Dat ^ P = x.y dieu ki^n Va^+b^+N/2ab = 8V2 >4P h f phuong trinh da ^|ia + b)" - 4ab(a + b)^ + 2a^b^ + yflab = 8V2 IS = a + b 19^ > 4P D^t -I „ , dieu ki?n 0 V - P - P +N/2P = 8N/2 o S = P = 4«.a = b = o x = y = Ngoai ta cung c6 the giai ngan gpn hon nhu sau: ^ ( x + y ) + ; ^ = 16 S->/P=3 S>3;P = ( S - f S + + 2VS + P + l = 2^S + ( S - f + = - S 4(52 + 8S +10) = 196 - 28S + Ta viet Igi h^ phuong a +b=4 S=4 cho tro thanh: 3£SXj=3;X2=-2 x + y + ^ / ^ = 8N/2 3 Bien doi phuong trinh (1): Vay h? da cho c6 nghif m (x; y) = (3; ) x+y ^ ' x+v * D§t x + y = S,xy = P ta CO phuong trinh: 5^+ 'J / 2P-1 = s o S + P - S P - S = o S ( S - l ) - P ( S - l ) = » ( S - l ) ( S + S - P ) = Cty TNHH Vi > 4P,S > suy +y X x= y = 0, y = x + y + l = l - x ^ - y ^ < = > x ^ + y ^ + x + y = ( k h o n g Vay he c6 n g h i ^ m : ( x ; y ) = ( l ; ) , ( ; l ) ^ x + y + —+ — = x y 1' X + — [ + y+ - = I yJ y + - X + — y x+— Dat -2P=9 S =5 oS = 5,P-6 « =9 x + - = 2;y + - = X y d) x = l;y = tuong duong vol : 3±S] M p t h? p h u o n g t r i n h an x , y dup-c gpi la d o i x u n g loai neu t r o n g h? = S =P X + — X phuong t r i n h ta d o ! v a i t r o x , y c h o t h i p h u o n g Tinh chat.: N e u (x,);yQ) la nghi^m cua h$ t h i ( y o ; x o ) c i j n g la n g h i ^ m + Phuong phap giai: T r u ve v o i ve hai p h u o n g t r i n h cua h ^ ta du^c m p t p h u o n g t r i n h c6 dang 3±75 (x-y)[f(x;y)] = « 3±V5 x =—-—;y =1 'x-y =0 f(x;y) = Ta cijng CO the d u n g p h u o n g phap ham so'de t i m quan he x = y V i di^ 1: G i a i cat h ^ phuong trinh sau: (3±^y5 a) xy (x + y ) ( x + y + x y ) = 30 (x-l)(y2+6) = y(x2+l x^ +v/x = y b) ( y - l ) ( x + ) = x(y2+l) x y ( x + y ) + x + y + xy = 11 x-' + X - + V x + = y Dat x y ( x + y ) = a;xy + X + y = b Ta thu du(?c h f : d) y^ + 3y - + ^ y + = x x + \ / x - x + =3^-^+1 xy(x + y) = ab = 30 a = 5;b = xy + a + b=l l a = 6;b = ' xy(x + y) = xy + X X a) Dieu k i ^ n : x,y > T r u hai p h u o n g t r i n h ciia h$ cho ta t h u dugc: x2 + V ; ^ _ ( y + ^ J ( y - x ) + y=5 « (N/^ - Vy ) [ ( ^ + 7y )(^ + y ) + + ( ^ + y ) ] = + y=3 xy(x + y) = X xy + xy = X + y=5 x + y =2 x = 2;y = l (L) x = l;y=2 + 2=3"-^+! Giai: +y- xy = THI: t r i n h t r o + x + — = 3;y + — = X y Vay h ^ da cho c6 n g h i ^ m : (x; y ) = 1; ± V T 5q:>/2T^ p h u o n g t r i n h tro thanh: S II) He DOI XLfNG LOAI da cho t u o n g d u o n g : ( ;y = - -;y = — ^ — Vay h# da cho c6 n g h i ^ m ( x ; y ) = ( l ; ) , ( - ; ) c) Dieu ki?n: \y ^0 : thoa i man dieu ki^n) X Vift + S - 2P > D o d o S = V o i X + y = thay vao (2) ta du