1 00:00:01,010 --> 00:00:04,520 Vitam vas na prednaske o kvadratickej rovnici. 2 00:00:04,520 --> 00:00:06,730 Taka kvadraticka rovnica, to znie ako nieco 3 00:00:06,730 --> 00:00:07,810 velmi zlozite. 4 00:00:07,810 --> 00:00:09,930 Ked skutocne prvykrat uvidite kvadraticku rovnicu, 5 00:00:09,930 --> 00:00:11,590 poviete si, nielenze to znie 6 00:00:11,590 --> 00:00:13,110 zlozito, ale to a jzlozite je. 7 00:00:13,110 --> 00:00:14,930 Nastastie vsak v priebehu tejto prednasky uvidite, 8 00:00:14,930 --> 00:00:16,580 ze to v skutocnosti nie je take tazke. 9 00:00:16,580 --> 00:00:19,040 V buducej prednaske vam ukazem, 10 00:00:19,040 --> 00:00:21,300 ako to bolo odvodene. 11 00:00:21,300 --> 00:00:24,810 Vo vseobecnosti ste sa uz naucili rozlozit 12 00:00:24,810 --> 00:00:25,810 rovnicu druheho stupna. 13 00:00:25,810 --> 00:00:30,910 Naucili ste sa, ze ak som mal, povedzme, x na druhu, 14 00:00:30,910 --> 00:00:40,340 minus x, minus 6, rovna sa 0. 15 00:00:40,340 --> 00:00:42,970 Ak by som mal taku rovnicu, x na druhu minus x minus x sa rovna 16 00:00:42,970 --> 00:00:48,720 nula, mohli by ste ju rozlozit ako x minus 3 a 17 00:00:48,720 --> 00:00:52,210 x plus 2 rovna sa 0. 18 00:00:52,210 --> 00:00:54,955 Staci, ak x minus 3 sa rovna 0, alebo 19 00:00:54,955 --> 00:00:57,073 x plus 2 sa rovna 0. 20 00:00:57,073 --> 00:01:03,512 Takze x minus 3 sa rovna 0 alebo x plus 2 sa rovna 0. 21 00:01:03,512 --> 00:01:08,500 Takze x sa rovna 3 alebo minus 2. 22 00:01:08,500 --> 00:01:17,980 Graficke zobrazenie tohto by bolo, ak by som mal 23 00:01:17,980 --> 00:01:26,150 funkciu f (x) sa rovna x na druhu minus x minus 6. 24 00:01:26,150 --> 00:01:28,760 Tato os je f osi x. 25 00:01:28,760 --> 00:01:32,670 Mozno ti je znamejsia os y, ale na ucely 26 00:01:32,670 --> 00:01:34,780 nasho problemu na tom nezalezi. 27 00:01:34,780 --> 00:01:36,270 Toto je os x. 28 00:01:36,270 --> 00:01:40,430 Ak by som chcel znazornit tuto rovnicu, x na druhu minus x, 29 00:01:40,430 --> 00:01:42,380 minus 6, vyzeralo by to asi takto. 30 00:01:42,380 --> 00:01:50,130 Trochu ako -- toto je f (x) rovna sa minus 6. 31 00:01:50,130 --> 00:01:52,900 Graf by vyzeral asi takto. 32 00:01:52,900 --> 00:01:57,150 Pojde to smerom hore. 33 00:02:00,030 --> 00:02:03,150 Vedz, ze to ide cez minus 6, pretoze ked sa x rovna 0, 34 00:02:03,150 --> 00:02:05,110 f (x) sa rovna minus 6. 35 00:02:05,110 --> 00:02:07,800 Takto ja viem, ze to ide cez tento bod. 36 00:02:07,800 --> 00:02:11,520 Viem aj, ze ked f(x) sa rovna 0, tak f(x) sa rovna 37 00:02:11,520 --> 00:02:14,960 0 pozdlz celej osi x. spravne? 38 00:02:14,960 --> 00:02:16,600 Tu je 1. 39 00:02:16,600 --> 00:02:17,870 Tu je 0. 40 00:02:17,870 --> 00:02:19,160 Tu je minus 1. 41 00:02:19,160 --> 00:02:21,510 Takze tu to je, kde f(x) sa rovna 0, na 42 00:02:21,510 --> 00:02:23,420 celej tejto osi x, spravne? 43 00:02:23,420 --> 00:02:29,210 Vieme, ze to sa rovna 0 v bodoch, kde x sa rovna 3 a 44 00:02:29,210 --> 00:02:32,330 x sa rovna minus 2. 45 00:02:32,330 --> 00:02:34,360 Toto je vlastne to, co sme tu riesili. 46 00:02:34,360 --> 00:02:36,440 Mozno ked sme sa venovali problemom s rozlozenim, 47 00:02:36,440 --> 00:02:38,940 neuvedomili sme si graficky, co robime. 48 00:02:38,940 --> 00:02:42,070 Ale ak sme si povedali, ze f(x) sa rovna tejto funkcii, 49 00:02:42,070 --> 00:02:43,270 prisudzujeme jej hodnotu nula. 50 00:02:43,270 --> 00:02:44,820 Hovorime tomu funkcia. Kedy sa 51 00:02:44,820 --> 00:02:48,220 tato funkcia rovna 0? 52 00:02:48,220 --> 00:02:49,390 kedy? 53 00:02:49,390 --> 00:02:51,720 Rovna sa nule v tychto bodoch, ano? 54 00:02:51,720 --> 00:02:55,360 Pretoze tu sa f(x) rovna 0. 55 00:02:55,360 --> 00:02:57,490 Ked sme toto vyriesili 56 00:02:57,490 --> 00:03:01,970 rozlozenim, prisli sme na to, ze hodnoty x, ktore tvorili f(x), 57 00:03:01,970 --> 00:03:04,160 sa rovnaju 0, co su tieto dva body. 58 00:03:04,160 --> 00:03:06,740 Teraz trocha terminologie - nazyvaju sa 59 00:03:06,740 --> 00:03:09,860 nulami, alebo aj korenmi f(x). 60 00:03:09,860 --> 00:03:12,470 Trocha si to zopakujme. 61 00:03:14,810 --> 00:03:23,700 Ak by som mal nieco ako f(x) sa rovna x na druhu plus 62 00:03:23,700 --> 00:03:29,550 4 krat x plus 4, a opytal by som sa ta, kde su nuly ci 63 00:03:29,550 --> 00:03:31,770 korene f(x)? 64 00:03:31,770 --> 00:03:33,970 To je to iste, ako opytat sa ta: kde f(x) 65 00:03:33,970 --> 00:03:36,300 pretina os x? 66 00:03:36,300 --> 00:03:38,210 Pretina ju, ked f(x) 67 00:03:38,210 --> 00:03:39,440 sa rovna 0, ano? 68 00:03:39,440 --> 00:03:42,120 Ak teda myslime graf, ktory som predtym nakreslil. 69 00:03:42,120 --> 00:03:45,720 Povedzme, ze f(x) sa rovna 0, potom mozeme 70 00:03:45,720 --> 00:03:51,860 povedat, ze 0 sa rovna x na druhu plus 4 krat x plus 4. 71 00:03:51,860 --> 00:03:53,940 Vieme, ze to mozeme rozlozit, teda x 72 00:03:53,940 --> 00:03:57,080 plus 2 krat x plus 2. 73 00:03:57,080 --> 00:04:07,090 Vieme, ze sa to rovna 0, ak sa x rovna minus 2. 74 00:04:07,090 --> 00:04:10,170 x sa rovna minus 2. 75 00:04:13,940 --> 00:04:18,270 No, toto je trocha preklep, takze x sa rovna minus 2. 76 00:04:18,270 --> 00:04:22,380 Tak teraz uz vieme, ako najdeme korene, ked sa urcita 77 00:04:22,380 --> 00:04:24,560 rovnica da lahko rozlozit. 78 00:04:24,560 --> 00:04:27,500 Ale skusme rovnicu, ktoru nie je v skutocnosti 79 00:04:27,500 --> 00:04:28,850 take lahke rozlozit. 80 00:04:28,850 --> 00:04:32,120 Priklad: mame f(x) sa rovna minus 10 krat x 81 00:04:39,750 --> 00:04:45,380 na druhu minus 9 krat x plus 1. 82 00:04:45,380 --> 00:04:47,580 Ked sa na to pozriem, aj keby som to vydelil 10, 83 00:04:47,580 --> 00:04:48,650 ostali by mi tu nejake zlomky. 84 00:04:48,650 --> 00:04:53,130 Je velmi tazke predstavit si rozlozenie tejto kvadratickej rovnice. 85 00:04:53,130 --> 00:04:54,860 Toto sa vlastne vola kvadraticka rovnica, alebo 86 00:04:54,860 --> 00:04:57,580 druhostupnovy polynomial. 87 00:04:57,580 --> 00:04:59,600 Skusime to vyriesit. 88 00:04:59,600 --> 00:05:02,420 Pretoze chceme zistit, kedy sa to rovna 0. 89 00:05:02,420 --> 00:05:07,130 Minus 10 krat x na druhu minus 9 krat x plus 1. 90 00:05:07,130 --> 00:05:09,090 Chceme zistit, ake hodnoty musi mat x, aby 91 00:05:09,090 --> 00:05:11,260 sa tato rovnica rovnala 0. 92 00:05:11,260 --> 00:05:13,730 A tu mozme pouzit pomocku nazvanu vzorec kvadratickej rovnice. 93 00:05:13,730 --> 00:05:15,625 Teraz vam dam jednu radu v matematike, 94 00:05:15,625 --> 00:05:18,030 ktoru je dobre si zapamatat. 95 00:05:18,030 --> 00:05:21,330 Korene kvadratickej rovnice sa vypocitaju podla daneho vzorca. 96 00:05:21,330 --> 00:05:24,810 Kvadraticka rovnica ma vo vseobecnosti takyto tvar: 97 00:05:24,810 --> 00:05:31,900 A krat x na druhu plus B krat x plus C sa rovna 0. 98 00:05:31,900 --> 00:05:35,790 V nasom priklade je A minus 10, 99 00:05:35,790 --> 00:05:39,940 B je minus 9, a C je 1. 100 00:05:39,940 --> 00:05:48,040 Vzorec je: korene x sa rovnaju minus B plus alebo minus 101 00:05:48,040 --> 00:05:58,060 druha odmocnina B na druhu minus 4 krat A krat C, 102 00:05:58,060 --> 00:06:00,230 vsetko to delene 2 krat A. 103 00:06:00,230 --> 00:06:02,843 Viem, ze to vyzera zlozito, ale cim viacej to budes pouzivat, 104 00:06:02,843 --> 00:06:04,400 uvidis, ze to v skutocnosti nie je az take zle. 105 00:06:04,400 --> 00:06:07,720 Je dobre si ten vzorec zapamatat. 106 00:06:07,720 --> 00:06:10,730 Aplikujme tento vzorec na nasu rovnicu, 107 00:06:10,730 --> 00:06:12,670 ktoru sme si napisali. 108 00:06:12,670 --> 00:06:15,260 Takze - pozri sa, A je iba koeficient 109 00:06:15,260 --> 00:06:18,610 clena x na druhu, ano? 110 00:06:18,610 --> 00:06:20,300 takze A je koeficient clena x na druhú. 111 00:06:20,300 --> 00:06:23,570 B je koeficient clena x. C je konštanta. 112 00:06:23,570 --> 00:06:25,100 Takze aplikujme tento vzorec na nasu rovnicu. 113 00:06:25,100 --> 00:06:26,250 Kolko je B? 114 00:06:26,250 --> 00:06:28,700 B je minus 9. 115 00:06:28,700 --> 00:06:29,970 Mozeme to vidiet tu. 116 00:06:29,970 --> 00:06:33,980 B je minus 9, A je minus 10. 117 00:06:33,980 --> 00:06:34,970 C je 1. 118 00:06:34,970 --> 00:06:36,090 Ano? 119 00:06:36,090 --> 00:06:42,350 Ak B je minus 9 - tak potom mame minus minus 9. 120 00:06:42,350 --> 00:06:49,260 Plus alebo mínus druhá odmocnina minus 9 na druhú. 121 00:06:49,260 --> 00:06:49,810 To je 81. 122 00:06:49,810 --> 00:06:53,140 Mínus 4 krát A. 123 00:06:56,940 --> 00:06:59,760 A je mínus 10. 124 00:06:59,760 --> 00:07:03,240 Mínus 10 krát C, ktore je 1. 125 00:07:03,240 --> 00:07:05,110 Viem, že je to chaoticke, ale dúfam, že to 126 00:07:05,110 --> 00:07:06,470 chapes. 127 00:07:06,470 --> 00:07:09,560 Všetko delene 2 krát A. 128 00:07:09,560 --> 00:07:14,050 A je mínus 10, takze 2 krát A je potom mínus 20. 129 00:07:14,050 --> 00:07:14,990 Tak si to zjednodušme. 130 00:07:14,990 --> 00:07:19,410 minus minus 9, to je kladne 9. 131 00:07:19,410 --> 00:07:26,460 Plus alebo mínus druhá odmocnina z 81. 132 00:07:26,460 --> 00:07:30,660 Máme minus 4 krat A, ktore je minus 10 . 133 00:07:30,660 --> 00:07:31,870 Tu je mínus 10. 134 00:07:31,870 --> 00:07:33,280 Viem, že je to veľmi komplikované, je mi to luto, 135 00:07:33,280 --> 00:07:34,380 krat C, teda krat 1. 136 00:07:34,380 --> 00:07:39,410 minus 4 krat minus 10 je 40, kladne 40. 137 00:07:39,410 --> 00:07:41,040 Kladne 40. 138 00:07:41,040 --> 00:07:46,070 To vsetko vydelime minus 20. . 139 00:07:46,070 --> 00:07:48,300 81 plus 40 je 121. 140 00:07:48,300 --> 00:07:52,330 9 plus alebo mínus druhá odmocnina 141 00:07:52,330 --> 00:07:58,290 zo 121 delene mínus 20. 142 00:07:58,290 --> 00:08:01,620 Druhá odmocnina zo 121 je 11. 143 00:08:01,620 --> 00:08:03,170 Pôjdem sem. 144 00:08:03,170 --> 00:08:06,184 Dúfam, že nestratís prehľad o tom, čo robím. 145 00:08:06,184 --> 00:08:13,720 9 plus alebo mínus 11, delene mínus 20. 146 00:08:13,720 --> 00:08:19,090 9 plus 11 delene mínus 20, to je 9 147 00:08:19,090 --> 00:08:22,540 plus 11 je 20, takže to je 20 delene mínus 20, 148 00:08:22,540 --> 00:08:23,730 co sa rovná minus 1 . 149 00:08:23,730 --> 00:08:24,900 Takže tu mame prvy koreň. 150 00:08:24,900 --> 00:08:28,260 To je 9 plus - pretože to je plus alebo mínus. 151 00:08:28,260 --> 00:08:33,790 A ten druhý koreň potom bude 9 mínus 11 delene minus 20, 152 00:08:33,790 --> 00:08:37,720 co sa rovná mínus 2 delene mínus 20, 153 00:08:37,720 --> 00:08:40,700 co sa rovná 1 lomene 10. 154 00:08:40,700 --> 00:08:42,690 Tak toto je dalsi koren. 155 00:08:42,690 --> 00:08:48,950 Ak by sme tuto rovnicu zobrazili na grafe, videli by sme, ze v 156 00:08:48,950 --> 00:08:52,640 bodoch minus 1 a 1/10 naozaj pretína os x. 157 00:08:52,640 --> 00:08:57,770 Alebo f ( x) sa rovna 0 v bodoch, kde x sa rovna 158 00:08:57,770 --> 00:09:01,690 minus 1 alebo x sa rovná 1/10. 159 00:09:01,690 --> 00:09:04,080 V časti 2 budu dalsie príklady, pretože si 160 00:09:04,080 --> 00:09:06,100 myslím, ze ak niečo, tak možno som ta 161 00:09:06,100 --> 00:09:08,120 tymto trocha doplietol. 162 00:09:08,120 --> 00:09:11,680 Uvidíme sa teda v časti 2 s dalsimi 163 00:09:11,680 --> 00:09:12,150 kvadratickymi rovnicami. 164 00:09:12,150 --> 00:09:14,083 ...