{"id":2133,"date":"2022-03-30T20:47:15","date_gmt":"2022-03-30T20:47:15","guid":{"rendered":"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/reseny-priklad-2-3\/"},"modified":"2022-03-31T19:30:08","modified_gmt":"2022-03-31T19:30:08","slug":"reseny-priklad-2-3","status":"publish","type":"lesson","link":"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/reseny-priklad-2-3\/","title":{"rendered":"\u0158e\u0161en\u00fd p\u0159\u00edklad 2"},"content":{"rendered":"

Zad\u00e1n\u00ed<\/strong>: \u0158e\u0161t\u011b v oboru R rovnici:<\/p>\n

$$\\sqrt{x+1}+\\sqrt{x+2}=\\sqrt{4x+5}$$<\/p>\n

 <\/p>\n

\u0158e\u0161en\u00ed<\/strong>:<\/p>\n

Celou rovnice umocn\u00edme:<\/p>\n

$$\\sqrt{x+1}+\\sqrt{x+2}=\\sqrt{4x+5}\/\u00b2$$<\/p>\n

$${(\\sqrt{x+1}+\\sqrt{x+2})}^{2}={(\\sqrt{4x+5})}^{2}$$<\/p>\n

M\u00e1me na pam\u011bti to, \u017ee v\u00fdraz na lev\u00e9 stran\u011b rovnice mus\u00edme umocnit podle vzorce:<\/p>\n

(a+b)\u00b2=a\u00b2+2ab+b\u00b2<\/em><\/strong>, kde\u00a0a=\u221ax+1<\/em><\/strong>\u00a0a\u00a0b=\u221ax+2<\/strong><\/em><\/p>\n

$$x+1+2\\sqrt{x+1}\\sqrt{x+2}+x+2=4x+5$$<\/p>\n

$$2x+3+2\\sqrt{x+1}\\sqrt{x+2}=4x+5$$<\/p>\n

$$-2+2\\sqrt{x+1}\\sqrt{x+2}=2x\/:2$$<\/p>\n

$$-1+\\sqrt{x+1}\\sqrt{x+2}=x\/\u00b2$$<\/p>\n

$${(-1)}^{2}+(x+1)(x+2)=x^{2}$$<\/p>\n

$$1+{x}^{2}+2x+x+2=x^{2}$$<\/p>\n

$$3x=-3$$<\/p>\n

$$x=-1$$<\/p>\n

Ne\u017e prohl\u00e1s\u00edme v\u00fdsledek za spr\u00e1vn\u00fd, mus\u00edme prov\u00e9st zkou\u0161ku<\/p>\n

$$L(-1)=\\sqrt{(-1)+1}+\\sqrt{(-1)+2}=\\sqrt{0}+\\sqrt{1}=1$$<\/p>\n

$$P(-1)=\\sqrt{4(-1)+5}=\\sqrt{1}=1$$<\/p>\n

$$L=P$$<\/p>\n

K={-1}<\/strong><\/em><\/p>\n","protected":false},"comment_status":"closed","ping_status":"closed","template":"","class_list":["post-2133","lesson","type-lesson","status-publish","hentry"],"yoast_head":"\n\u0158e\u0161en\u00fd p\u0159\u00edklad 2 | Studijni-svet.cz<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/reseny-priklad-2-3\/\" \/>\n<meta property=\"og:locale\" content=\"cs_CZ\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u0158e\u0161en\u00fd p\u0159\u00edklad 2 | Studijni-svet.cz\" \/>\n<meta property=\"og:description\" content=\"Zad\u00e1n\u00ed: \u0158e\u0161t\u011b v oboru R rovnici: $$sqrt{x+1}+sqrt{x+2}=sqrt{4x+5}$$   \u0158e\u0161en\u00ed: Celou rovnice umocn\u00edme: $$sqrt{x+1}+sqrt{x+2}=sqrt{4x+5}\/\u00b2$$ $${(sqrt{x+1}+sqrt{x+2})}^{2}={(sqrt{4x+5})}^{2}$$ M\u00e1me na pam\u011bti to, \u017ee v\u00fdraz na lev\u00e9 stran\u011b rovnice mus\u00edme umocnit podle vzorce: (a+b)\u00b2=a\u00b2+2ab+b\u00b2, kde\u00a0a=\u221ax+1\u00a0a\u00a0b=\u221ax+2 $$x+1+2sqrt{x+1}sqrt{x+2}+x+2=4x+5$$ $$2x+3+2sqrt{x+1}sqrt{x+2}=4x+5$$ $$-2+2sqrt{x+1}sqrt{x+2}=2x\/:2$$ $$-1+sqrt{x+1}sqrt{x+2}=x\/\u00b2$$ $${(-1)}^{2}+(x+1)(x+2)=x^{2}$$ $$1+{x}^{2}+2x+x+2=x^{2}$$ $$3x=-3$$ $$x=-1$$ Ne\u017e prohl\u00e1s\u00edme v\u00fdsledek za spr\u00e1vn\u00fd, mus\u00edme prov\u00e9st zkou\u0161ku $$L(-1)=sqrt{(-1)+1}+sqrt{(-1)+2}=sqrt{0}+sqrt{1}=1$$ $$P(-1)=sqrt{4(-1)+5}=sqrt{1}=1$$ $$L=P$$ K={-1}\" \/>\n<meta property=\"og:url\" content=\"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/reseny-priklad-2-3\/\" \/>\n<meta property=\"og:site_name\" content=\"Studijni-svet.cz\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/studijnisvet\/\" \/>\n<meta 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