{"id":20496,"date":"2021-12-17T13:31:52","date_gmt":"2021-12-17T06:31:52","guid":{"rendered":"https:\/\/www.vietlearn.org\/post\/?p=20496"},"modified":"2021-12-17T13:31:54","modified_gmt":"2021-12-17T06:31:54","slug":"mot-so-ki-nang-giai-he-phuong-trinh","status":"publish","type":"post","link":"https:\/\/www.vietlearn.org\/kien-thuc\/mot-so-ki-nang-giai-he-phuong-trinh\/","title":{"rendered":"M\u1ed8T S\u1ed0 K\u0128 N\u0102NG GI\u1ea2I H\u1ec6 PH\u01af\u01a0NG TR\u00ccNH"},"content":{"rendered":"

Trong c\u00e1c \u0111\u1ec1 thi \u0111\u1ea1i h\u1ecdc nh\u1eefng n\u0103m g\u1ea7n \u0111\u00e2y , ta g\u1eb7p r\u1ea5t nhi\u1ec1u b\u00e0i to\u00e1n v\u1ec1 h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh . Nh\u1eb1m gi\u00fap c\u00e1c b\u1ea1n \u00f4n thi<\/a> t\u1ed1t , b\u00e0i vi\u1ebft n\u00e0y t\u00f4i xin gi\u1edbi thi\u1ec7u m\u1ed9t s\u1ed1 d\u1ea1ng b\u00e0i v\u00e0 k\u0129 n\u0103ng gi\u1ea3i ch\u00fang<\/p>\n

I.H\u1ec6 S\u1eec D\u1ee4NG PH\u01af\u01a0NG PH\u00c1P BI\u1ebeN \u0110\u1ed4I T\u01af\u01a0NG \u0110\u01af\u01a0NG.<\/strong><\/p>\n

\u0110\u1eb7c \u0111i\u1ec3m chung c\u1ee7a d\u1ea1ng h\u1ec7 n\u00e0y l\u00e0 s\u1eed d\u1ee5ng c\u00e1c k\u0129 n\u0103ng bi\u1ebfn \u0111\u1ed5i \u0111\u1ed3ng nh\u1ea5t \u0111\u1eb7c bi\u1ec7t l\u00e0 k\u0129 n\u0103ng ph\u00e2n t\u00edch nh\u1eb1m \u0111\u01b0a m\u1ed9t PT trong h\u1ec7 v\u1ec1 d\u1ea1ng \u0111\u01a1n gi\u1ea3n ( c\u00f3 th\u1ec3 r\u00fat theo y ho\u1eb7c ng\u01b0\u1ee3c l\u1ea1i ) r\u1ed3i th\u1ebf v\u00e0o PT c\u00f2n l\u1ea1i trong h\u1ec7 .<\/p>\n

*Lo\u1ea1i th\u1ee9 nh\u1ea5t , trong h\u1ec7 c\u00f3 m\u1ed9t ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc nh\u1ea5t v\u1edbi \u1ea9n x ho\u1eb7c y khi \u0111\u00f3 ta t\u00ecm c\u00e1ch r\u00fat y theo x ho\u1eb7c ng\u01b0\u1ee3c l\u1ea1i<\/em><\/p>\n

V\u00ed d\u1ee5 1 .<\/strong> Gi\u1ea3i h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh \"IMG_256\"<\/p>\n

Gi\u1ea3i.<\/p>\n

D\u1ec5 th\u1ea5y x = 0 kh\u00f4ng th\u1ecfa m\u00e3n PT(2) n\u00ean t\u1eeb (2) ta c\u00f3 : \"IMG_257\" thay v\u00e0o (1) ta \u0111\u01b0\u1ee3c<\/p>\n

\"IMG_258\"<\/p>\n

\"IMG_259\" \"IMG_260\"<\/p>\n

T\u1eeb \u0111\u00f3 , ta \u0111\u01b0\u1ee3c c\u00e1c nghi\u1ec7m c\u1ee7a h\u1ec7 l\u00e0 : (1;-1) , (-2; \"IMG_261\" )<\/p>\n

*Lo\u1ea1i th\u1ee9 hai , M\u1ed9t ph\u01b0\u01a1ng tr\u00ecnh trong h\u1ec7 c\u00f3 th\u1ec3 \u0111\u01b0a v\u1ec1 d\u1ea1ng t\u00edch c\u1ee7a c\u00e1c ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc nh\u1ea5t hai \u1ea9n<\/em><\/p>\n

V\u00ed d\u1ee5 2 .<\/strong> Gi\u1ea3i h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh \"IMG_262\"<\/p>\n

Gi\u1ea3i .<\/p>\n

\u0110i\u1ec1u ki\u1ec7n : x\u22651 ; y\u22650<\/p>\n

PT (1) \"IMG_263\" ( t\u1eeb \u0111i\u1ec1u ki\u1ec7n ta c\u00f3 x+y>0)<\/p>\n

\"IMG_264\" thay v\u00e0o PT (2) ta \u0111\u01b0\u1ee3c :<\/p>\n

\"IMG_265\"<\/p>\n

*lo\u1ea1i th\u1ee9 ba , \u0111\u01b0a m\u1ed9t ph\u01b0\u01a1ng tr\u00ecnh trong h\u1ec7 v\u1ec1 d\u1ea1ng ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc hai c\u1ee7a m\u1ed9t \u1ea9n , \u1ea9n c\u00f2n l\u1ea1i l\u00e0 tham s\u1ed1<\/em><\/p>\n

V\u00ed d\u1ee5 3.<\/strong> Gi\u1ea3i h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh \"IMG_266\"<\/p>\n

Gi\u1ea3i .<\/p>\n

Bi\u1ebfn \u0111\u1ed5i PT (2) v\u1ec1 d\u1ea1ng \"IMG_267\"<\/p>\n

Coi PT (2) l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh \u1ea9n y tham s\u1ed1 x ta c\u00f3 \"IMG_268\" t\u1eeb \u0111\u00f3 ta \u0111\u01b0\u1ee3c nghi\u1ec7m \"IMG_269\"<\/p>\n

Thay (3) v\u00e0o (1) ta \u0111\u01b0\u1ee3c : \"IMG_270\"<\/p>\n

Thay (4) v\u00e0o (1) ta \u0111\u01b0\u1ee3c : \"IMG_271\"<\/p>\n

V\u1eady nghi\u1ec7m c\u1ee7a h\u1ec7 l\u00e0 : (0;4) , (4;0) , ( \"IMG_272\" ;0)<\/p>\n

II.H\u1ec6 S\u1eec D\u1ee4NG PH\u01af\u01a0NG PH\u00c1P \u0110\u1eb6T \u1ea8N PH\u1ee4<\/strong><\/p>\n

\u0110i\u1ec3m quan tr\u1ecdng nh\u1ea5t trong h\u1ec7 d\u1ea1ng n\u00e0y l\u00e0 ph\u00e1t hi\u1ec7n \u1ea9n ph\u1ee5 \"IMG_273\" c\u00f3 ngay trong t\u1eebng ph\u01b0\u01a1ng tr\u00ecnh ho\u1eb7c xu\u1ea5t hi\u1ec7n sau m\u1ed9t ph\u00e9p bi\u1ebfn \u0111\u1ed5i h\u1eb1ng \u0111\u1eb3ng th\u1ee9c c\u01a1 b\u1ea3n ho\u1eb7c ph\u00e9p chia cho m\u1ed9t bi\u1ec3u th\u1ee9c kh\u00e1c 0.<\/p>\n

V\u00ed d\u1ee5 4<\/strong>. Gi\u1ea3i h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh \"IMG_274\"<\/p>\n

Gi\u1ea3i .<\/p>\n

D\u1ec5 th\u1ea5y y=0 kh\u00f4ng th\u1ecfa m\u00e3n PT(1) n\u00ean HPT \"IMG_275\"<\/p>\n

\u0110\u1eb7t \"IMG_276\" gi\u1ea3i h\u1ec7 ta \u0111\u01b0\u1ee3c a=b=1 t\u1eeb \u0111\u00f3 ta c\u00f3 h\u1ec7 \"IMG_277\"<\/p>\n

H\u1ec7 n\u00e0y b\u1ea1n \u0111\u1ecdc c\u00f3 th\u1ec3 gi\u1ea3i d\u1ec5 d\u00e0ng.<\/p>\n

V\u00ed d\u1ee5 5.<\/strong> Gi\u1ea3i h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh \"IMG_278\"<\/p>\n

Gi\u1ea3i . \u0110i\u1ec1u ki\u1ec7n : x +y \u22600<\/p>\n

HPT \"IMG_279\"<\/p>\n

\u0110\u1eb7t \"IMG_280\" ta \u0111\u01b0\u1ee3c h\u1ec7 \"IMG_281\"<\/p>\n

Gi\u1ea3i h\u1ec7 ta \u0111\u01b0\u1ee3c a=2 , b=1 ( do |a|\u22652 ) t\u1eeb \u0111\u00f3 ta c\u00f3 h\u1ec7 \"IMG_282\"<\/p>\n

III.H\u1ec6 S\u1eec D\u1ee4NG PH\u01af\u01a0NG PH\u00c1P H\u00c0M S\u1ed0<\/strong><\/p>\n

H\u1ec7 lo\u1ea1i n\u00e0y ta g\u1eb7p nhi\u1ec1u \u1edf hai d\u1ea1ng f(x)=0 (1)v\u00e0 f(x)=f(y) (2) v\u1edbi f l\u00e0 h\u00e0m \u0111\u01a1n \u0111i\u1ec7u tr\u00ean t\u1eadp D v\u00e0 x,y thu\u1ed9c D .Nhi\u1ec1u khi ta c\u1ea7n ph\u1ea3i \u0111\u00e1nh gi\u00e1 \u1ea9n x,y \u0111\u1ec3 x,y thu\u1ed9c t\u1eadp m\u00e0 h\u00e0m f \u0111\u01a1n \u0111i\u1ec7u<\/p>\n

* Lo\u1ea1i th\u1ee9 nh\u1ea5t , m\u1ed9t ph\u01b0\u01a1ng tr\u00ecnh trong h\u1ec7 c\u00f3 d\u1ea1ng f(x)=f(y) , ph\u01b0\u01a1ng tr\u00ecnh c\u00f2n l\u1ea1i gi\u00fap ta gi\u1edbi h\u1ea1n x,y thu\u1ed9c t\u1eadp D \u0111\u1ec3 tr\u00ean \u0111\u1ec3 tr\u00ean \u0111\u00f3 h\u00e0m f \u0111\u01a1n \u0111i\u1ec7u<\/em><\/p>\n

V\u00ed d\u1ee5 6 .<\/strong> Gi\u1ea3i h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh \"IMG_283\"<\/p>\n

Gi\u1ea3i .<\/p>\n

T\u1eeb PT (2) ta c\u00f3 \"IMG_284\"<\/p>\n

X\u00e9t h\u00e0m s\u1ed1 \"IMG_285\" c\u00f3 \"IMG_286\" do \u0111\u00f3 f(t) ngh\u1ecbch bi\u1ebfn tr\u00ean<\/p>\n

kho\u1ea3ng (-1;1) hay PT (1) \"IMG_287\" thay v\u00e0o PT (2) ta \u0111\u01b0\u1ee3c PT : \"IMG_288\"<\/p>\n

\u0110\u1eb7t a=x4<\/sup> \u22650 v\u00e0 gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh ta \u0111\u01b0\u1ee3c \"IMG_289\"<\/p>\n

*lo\u1ea1i th\u1ee9 hai , l\u00e0 d\u1ea1ng h\u1ec7 \u0111\u1ed1i x\u1ee9ng lo\u1ea1i hai m\u00e0 khi gi\u1ea3i th\u01b0\u1eddng d\u1eabn \u0111\u1ebfn c\u1ea3 hai tr\u01b0\u1eddng h\u1ee3p (1) v\u00e0 (2)<\/em><\/p>\n

V\u00ed d\u1ee5 7.<\/strong> Gi\u1ea3i h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh \"IMG_290\"<\/p>\n

Gi\u1ea3i .<\/p>\n

\u0110\u1eb7t \"IMG_291\" ta \u0111\u01b0\u1ee3c h\u1ec7 \"IMG_292\"<\/p>\n

Tr\u1eeb v\u1ebf v\u1edbi v\u1ebf 2 PT ta \u0111\u01b0\u1ee3c : \"IMG_293\" (3)<\/p>\n

X\u00e9t h\u00e0m s\u1ed1 \"IMG_294\"<\/p>\n

V\u00ec \"IMG_295\" do \u0111\u00f3 h\u00e0m s\u1ed1 f(t) \u0111\u1ed3ng bi\u1ebfn tr\u00ean R<\/p>\n

N\u00ean PT (3) \"IMG_296\" thay v\u00e0o PT (1) ta \u0111\u01b0\u1ee3c \"IMG_297\" (4)<\/p>\n

Theo nh\u1eadn x\u00e9t tr\u00ean th\u00ec \"IMG_298\" n\u00ean PT (4) \"IMG_299\" ( l\u1ea5y ln hai v\u1ebf )<\/p>\n

X\u00e9t h\u00e0m s\u1ed1 \"IMG_300\"<\/p>\n

hay h\u00e0m g(a) ngh\u1ecbch bi\u1ebfn tr\u00ean R v\u00e0 do PT (4) c\u00f3 nghi\u1ec7m a=0 n\u00ean PT (4) c\u00f3 nghi\u1ec7m duy nh\u1ea5t a=0<\/p>\n

T\u1eeb \u0111\u00f3 ta \u0111\u01b0\u1ee3c nghi\u1ec7m c\u1ee7a h\u1ec7 ban \u0111\u1ea7u l\u00e0 : x=y=1<\/p>\n

IV.H\u1ec6 S\u1eec D\u1ee4NG PH\u01af\u01a0NG PH\u00c1P \u0110\u00c1NH GI\u00c1<\/strong><\/p>\n

V\u1edbi ph\u01b0\u01a1ng ph\u00e1p n\u00e0y, c\u1ea7n l\u01b0u \u00fd ph\u00e1t hi\u1ec7n c\u00e1c bi\u1ec3u th\u1ee9c kh\u00f4ng \u00e2m v\u00e0 n\u1eafm v\u1eefng c\u00e1ch v\u1eadn d\u1ee5ng c\u00e1c b\u1ea5t \u0111\u1eb3ng th\u1ee9c c\u01a1 b\u1ea3n<\/p>\n

V\u00ed d\u1ee5 8<\/strong> . Gi\u1ea3i h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh \"IMG_301\"<\/p>\n

Gi\u1ea3i.<\/p>\n

C\u1ed9ng v\u1ebf v\u1edbi v\u1ebf hai PT ta \u0111\u01b0\u1ee3c \"IMG_302\" (1)<\/p>\n

Ta c\u00f3 : \"IMG_303\"<\/p>\n

T\u01b0\u01a1ng t\u1ef1 \"IMG_304\" m\u00e0 theo b\u1ea5t \u0111\u1eb3ng th\u1ee9c C\u00f4si \"IMG_305\" n\u00ean VT(1)\u2264VP(1)<\/p>\n

D\u1ea5u b\u1eb1ng x\u1ea3y ra khi \"IMG_306\" th\u1eed l\u1ea1i ta \u0111\u01b0\u1ee3c nghi\u1ec7m c\u1ee7a h\u1ec7 l\u00e0 : (0;0) , (1;1)<\/p>\n

V\u00ed d\u1ee5 9<\/strong> . Gi\u1ea3i h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh \"IMG_307\"<\/p>\n

Gi\u1ea3i.<\/p>\n

HPT \"IMG_308\"<\/p>\n

N\u1ebfu x>2 t\u1eeb (1) suy ra y-2<0 di\u1ec1u n\u00e0y m\u00e2u thu\u1eabn v\u1edbi PT(2) c\u00f3 (x-2) v\u00e0 (y-2) c\u00f9ng d\u1ea5u<\/p>\n

T\u01b0\u01a1ng t\u1ef1 v\u1edbi x<2 ta c\u0169ng suy ra \u0111i\u1ec1u v\u00f4 l\u00ed . V\u1eady nghi\u1ec7m c\u1ee7a h\u1ec7 l\u00e0 x=y=2<\/p>\n

Hy v\u1ecdng m\u1ed9t s\u1ed1 v\u00ed d\u1ee5 tr\u00ean s\u1ebd gi\u00fap b\u1ea1n ph\u1ea7n n\u00e0o k\u0129 n\u0103ng gi\u1ea3i h\u1ec7 .\u0110\u1ec3 k\u1ebft th\u00fac b\u00e0i vi\u1ebft m\u1eddi c\u00e1c b\u1ea1n c\u00f9ng gi\u1ea3i c\u00e1c h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh sau<\/p>\n

\"IMG_309\"<\/p>\n

\"IMG_310\"<\/p>\n","protected":false},"excerpt":{"rendered":"

Trong c\u00e1c \u0111\u1ec1 thi \u0111\u1ea1i h\u1ecdc nh\u1eefng n\u0103m g\u1ea7n \u0111\u00e2y , ta g\u1eb7p r\u1ea5t nhi\u1ec1u b\u00e0i to\u00e1n v\u1ec1 h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh . Nh\u1eb1m gi\u00fap c\u00e1c b\u1ea1n \u00f4n thi t\u1ed1t , b\u00e0i vi\u1ebft n\u00e0y t\u00f4i xin gi\u1edbi thi\u1ec7u m\u1ed9t s\u1ed1 d\u1ea1ng b\u00e0i v\u00e0 k\u0129 n\u0103ng gi\u1ea3i ch\u00fang I.H\u1ec6 S\u1eec D\u1ee4NG PH\u01af\u01a0NG PH\u00c1P BI\u1ebeN \u0110\u1ed4I T\u01af\u01a0NG \u0110\u01af\u01a0NG. […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_disable_autopaging":false},"categories":[41],"tags":[],"class_list":["post-20496","post","type-post","status-publish","format-standard","hentry","category-toan-lop-10"],"_links":{"self":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/posts\/20496","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/comments?post=20496"}],"version-history":[{"count":1,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/posts\/20496\/revisions"}],"predecessor-version":[{"id":20552,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/posts\/20496\/revisions\/20552"}],"wp:attachment":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/media?parent=20496"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/categories?post=20496"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/tags?post=20496"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}