Ci\u0105g geometryczny \u2013 obja\u015bnienie<\/title>\n<\/head>\n<body style=\"font-family: Arial; line-height: 1.6;\">\n\nAby trzy kolejne wyrazy tworzy\u0142y ci\u0105g geometryczny, stosunek drugiego do pierwszego musi by\u0107 r\u00f3wny stosunkowi trzeciego do drugiego:<\/p>\n\n\n (4x + 2) \/ (x + 2) = (x + 11) \/ (4x + 2)\n<\/p>\n\nNast\u0119pnie mno\u017cymy na krzy\u017c:<\/p>\n\n\n (4x + 2)\u00b2 = (x + 2)(x + 11)\n<\/p>\n\nRozwijamy obie strony:<\/p>\n\nLewa strona: 16x\u00b2 + 16x + 4<\/p>\nPrawa strona: x\u00b2 + 13x + 22<\/p>\n\nPrzenosimy wszystko na jedn\u0105 stron\u0119:<\/p>\n\n\n 15x\u00b2 + 3x \u2212 18 = 0\n<\/p>\n\nDzielimy ca\u0142e r\u00f3wnanie przez 3:<\/p>\n\n\n 5x\u00b2 + x \u2212 6 = 0\n<\/p>\n\nLiczymy delt\u0119:<\/p>\n\n\u0394 = 1 + 120 = 121, \u221a\u0394 = 11<\/p>\n\nObliczamy pierwiastki:<\/p>\n\nx\u2081 = (\u22121 \u2212 11) \/ 10 = \u22126\/5<\/p>\nx\u2082 = (\u22121 + 11) \/ 10 = 1<\/p>\n\n\n Ci\u0105g jest geometryczny dla: x = \u22126\/5 oraz x = 1<\/strong>\n<\/p>\n\n<hr>\n\n<h3>Przyznaj sobie punkty<\/h3>\n\n0 pkt<\/strong> \u2013 brak poprawnych krok\u00f3w.<\/p>\n1 pkt<\/strong> \u2013 cz\u0119\u015b\u0107 oblicze\u0144 wykonana poprawnie, ale nie ca\u0142o\u015b\u0107.<\/p>\n2 pkt<\/strong> \u2013 poprawne utworzenie r\u00f3wnania, przekszta\u0142cenia i obliczenie obu warto\u015bci x.<\/p>\n\n<\/body>\n<\/html>\n <\/div>\n <\/div>\n \n<\/div>\n<\/div>\n\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"zadanie 30 \u2013 czerwiec 2020 (zadanie otwarte) (2 pkt) Dany jest trzywyrazowy ci\u0105g (x+2, 4x+2, x+11). Oblicz te wszystkie warto\u015bci x, dla kt\u00f3rych ten ci\u0105g jest geometryczny. Przyznaj sobie samodzielnie punkty, zgodnie z proponowan\u0105 punktacj\u0105 (0 pkt, 1 pkt, 2 pkt)<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"saved_in_kubio":false,"_uag_custom_page_level_css":"","_themeisle_gutenberg_block_has_review":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[31,38,115,81,83],"tags":[29,118,112],"class_list":["post-945","post","type-post","status-publish","format-standard","hentry","category-ciag-geometryczny","category-czerwiec-2020","category-trzy-kolejne-wyrazy-ciagu","category-zadania-maturalne","category-zadania-tematami","tag-ciag-geometryczny","tag-czerwiec2020","tag-trzy-kolejne-wyrazy-ciagu"],"jetpack_featured_media_url":"","uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"1536x1536":false,"2048x2048":false,"kubio-fullhd":false,"woocommerce_thumbnail":false,"woocommerce_single":false,"woocommerce_gallery_thumbnail":false},"uagb_author_info":{"display_name":"adminmymatma","author_link":"https:\/\/mymatma.pl\/?author=1"},"uagb_comment_info":0,"uagb_excerpt":"zadanie 30 \u2013 czerwiec 2020 (zadanie otwarte) (2 pkt) Dany jest trzywyrazowy ci\u0105g (x+2, 4x+2, x+11). Oblicz te wszystkie warto\u015bci x, dla kt\u00f3rych ten ci\u0105g jest geometryczny. Przyznaj sobie samodzielnie punkty, zgodnie z proponowan\u0105 punktacj\u0105 (0 pkt, 1 pkt, 2 pkt)","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/945","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=945"}],"version-history":[{"count":4,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/945\/revisions"}],"predecessor-version":[{"id":979,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/945\/revisions\/979"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=945"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=945"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=945"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}

\n \n\n\n\n \n Ci\u0105g geometryczny \u2013 obja\u015bnienie<\/title>\n<\/head>\n<body style=\"font-family: Arial; line-height: 1.6;\">\n\nAby trzy kolejne wyrazy tworzy\u0142y ci\u0105g geometryczny, stosunek drugiego do pierwszego musi by\u0107 r\u00f3wny stosunkowi trzeciego do drugiego:<\/p>\n\n\n (4x + 2) \/ (x + 2) = (x + 11) \/ (4x + 2)\n<\/p>\n\nNast\u0119pnie mno\u017cymy na krzy\u017c:<\/p>\n\n\n (4x + 2)\u00b2 = (x + 2)(x + 11)\n<\/p>\n\nRozwijamy obie strony:<\/p>\n\nLewa strona: 16x\u00b2 + 16x + 4<\/p>\nPrawa strona: x\u00b2 + 13x + 22<\/p>\n\nPrzenosimy wszystko na jedn\u0105 stron\u0119:<\/p>\n\n\n 15x\u00b2 + 3x \u2212 18 = 0\n<\/p>\n\nDzielimy ca\u0142e r\u00f3wnanie przez 3:<\/p>\n\n\n 5x\u00b2 + x \u2212 6 = 0\n<\/p>\n\nLiczymy delt\u0119:<\/p>\n\n\u0394 = 1 + 120 = 121, \u221a\u0394 = 11<\/p>\n\nObliczamy pierwiastki:<\/p>\n\nx\u2081 = (\u22121 \u2212 11) \/ 10 = \u22126\/5<\/p>\nx\u2082 = (\u22121 + 11) \/ 10 = 1<\/p>\n\n\n Ci\u0105g jest geometryczny dla: x = \u22126\/5 oraz x = 1<\/strong>\n<\/p>\n\n<hr>\n\n<h3>Przyznaj sobie punkty<\/h3>\n\n0 pkt<\/strong> \u2013 brak poprawnych krok\u00f3w.<\/p>\n1 pkt<\/strong> \u2013 cz\u0119\u015b\u0107 oblicze\u0144 wykonana poprawnie, ale nie ca\u0142o\u015b\u0107.<\/p>\n2 pkt<\/strong> \u2013 poprawne utworzenie r\u00f3wnania, przekszta\u0142cenia i obliczenie obu warto\u015bci x.<\/p>\n\n<\/body>\n<\/html>\n <\/div>\n <\/div>\n \n<\/div>\n<\/div>\n\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"zadanie 30 \u2013 czerwiec 2020 (zadanie otwarte) (2 pkt) Dany jest trzywyrazowy ci\u0105g (x+2, 4x+2, x+11). Oblicz te wszystkie warto\u015bci x, dla kt\u00f3rych ten ci\u0105g jest geometryczny. Przyznaj sobie samodzielnie punkty, zgodnie z proponowan\u0105 punktacj\u0105 (0 pkt, 1 pkt, 2 pkt)<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"saved_in_kubio":false,"_uag_custom_page_level_css":"","_themeisle_gutenberg_block_has_review":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[31,38,115,81,83],"tags":[29,118,112],"class_list":["post-945","post","type-post","status-publish","format-standard","hentry","category-ciag-geometryczny","category-czerwiec-2020","category-trzy-kolejne-wyrazy-ciagu","category-zadania-maturalne","category-zadania-tematami","tag-ciag-geometryczny","tag-czerwiec2020","tag-trzy-kolejne-wyrazy-ciagu"],"jetpack_featured_media_url":"","uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"1536x1536":false,"2048x2048":false,"kubio-fullhd":false,"woocommerce_thumbnail":false,"woocommerce_single":false,"woocommerce_gallery_thumbnail":false},"uagb_author_info":{"display_name":"adminmymatma","author_link":"https:\/\/mymatma.pl\/?author=1"},"uagb_comment_info":0,"uagb_excerpt":"zadanie 30 \u2013 czerwiec 2020 (zadanie otwarte) (2 pkt) Dany jest trzywyrazowy ci\u0105g (x+2, 4x+2, x+11). Oblicz te wszystkie warto\u015bci x, dla kt\u00f3rych ten ci\u0105g jest geometryczny. Przyznaj sobie samodzielnie punkty, zgodnie z proponowan\u0105 punktacj\u0105 (0 pkt, 1 pkt, 2 pkt)","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/945","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=945"}],"version-history":[{"count":4,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/945\/revisions"}],"predecessor-version":[{"id":979,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/945\/revisions\/979"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=945"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=945"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=945"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}