WEBVTT 00:00:01.010 --> 00:00:04.520 Vitam vas na prednaske o kvadratickej rovnici. 00:00:04.520 --> 00:00:06.730 Taka kvadraticka rovnica, to znie ako nieco 00:00:06.730 --> 00:00:07.810 velmi zlozite. 00:00:07.810 --> 00:00:09.930 Ked skutocne prvykrat uvidite kvadraticku rovnicu, 00:00:09.930 --> 00:00:11.590 poviete si, nielenze to znie 00:00:11.590 --> 00:00:13.110 zlozito, ale to a jzlozite je. 00:00:13.110 --> 00:00:14.930 Nastastie vsak v priebehu tejto prednasky uvidite, 00:00:14.930 --> 00:00:16.580 ze to v skutocnosti nie je take tazke. 00:00:16.580 --> 00:00:19.040 V buducej prednaske vam ukazem, 00:00:19.040 --> 00:00:21.300 ako to bolo odvodene. 00:00:21.300 --> 00:00:24.810 Vo vseobecnosti ste sa uz naucili rozlozit 00:00:24.810 --> 00:00:25.810 rovnicu druheho stupna. 00:00:25.810 --> 00:00:30.910 Naucili ste sa, ze ak som mal, povedzme, x na druhu, 00:00:30.910 --> 00:00:40.340 minus x, minus 6, rovna sa 0. 00:00:40.340 --> 00:00:42.970 Ak by som mal taku rovnicu, x na druhu minus x minus x sa rovna 00:00:42.970 --> 00:00:48.720 nula, mohli by ste ju rozlozit ako x minus 3 a 00:00:48.720 --> 00:00:52.210 x plus 2 rovna sa 0. 00:00:52.210 --> 00:00:54.955 Staci, ak x minus 3 sa rovna 0, alebo 00:00:54.955 --> 00:00:57.073 x plus 2 sa rovna 0. 00:00:57.073 --> 00:01:03.512 Takze x minus 3 sa rovna 0 alebo x plus 2 sa rovna 0. 00:01:03.512 --> 00:01:08.500 Takze x sa rovna 3 alebo minus 2. 00:01:08.500 --> 00:01:17.980 Graficke zobrazenie tohto by bolo, ak by som mal 00:01:17.980 --> 00:01:26.150 funkciu f (x) sa rovna x na druhu minus x minus 6. 00:01:26.150 --> 00:01:28.760 Tato os je f osi x. 00:01:28.760 --> 00:01:32.670 Mozno ti je znamejsia os y, ale na ucely 00:01:32.670 --> 00:01:34.780 nasho problemu na tom nezalezi. 00:01:34.780 --> 00:01:36.270 Toto je os x. 00:01:36.270 --> 00:01:40.430 Ak by som chcel znazornit tuto rovnicu, x na druhu minus x, 00:01:40.430 --> 00:01:42.380 minus 6, vyzeralo by to asi takto. 00:01:42.380 --> 00:01:50.130 Trochu ako -- toto je f (x) rovna sa minus 6. 00:01:50.130 --> 00:01:52.900 Graf by vyzeral asi takto. 00:01:52.900 --> 00:01:57.150 Pojde to smerom hore. 00:02:00.030 --> 00:02:03.150 Vedz, ze to ide cez minus 6, pretoze ked sa x rovna 0, 00:02:03.150 --> 00:02:05.110 f (x) sa rovna minus 6. 00:02:05.110 --> 00:02:07.800 Takto ja viem, ze to ide cez tento bod. 00:02:07.800 --> 00:02:11.520 Viem aj, ze ked f(x) sa rovna 0, tak f(x) sa rovna 00:02:11.520 --> 00:02:14.960 0 pozdlz celej osi x. spravne? 00:02:14.960 --> 00:02:16.600 Tu je 1. 00:02:16.600 --> 00:02:17.870 Tu je 0. 00:02:17.870 --> 00:02:19.160 Tu je minus 1. 00:02:19.160 --> 00:02:21.510 Takze tu to je, kde f(x) sa rovna 0, na 00:02:21.510 --> 00:02:23.420 celej tejto osi x, spravne? 00:02:23.420 --> 00:02:29.210 Vieme, ze to sa rovna 0 v bodoch, kde x sa rovna 3 a 00:02:29.210 --> 00:02:32.330 x sa rovna minus 2. 00:02:32.330 --> 00:02:34.360 Toto je vlastne to, co sme tu riesili. 00:02:34.360 --> 00:02:36.440 Mozno ked sme sa venovali problemom s rozlozenim, 00:02:36.440 --> 00:02:38.940 neuvedomili sme si graficky, co robime. 00:02:38.940 --> 00:02:42.070 Ale ak sme si povedali, ze f(x) sa rovna tejto funkcii, 00:02:42.070 --> 00:02:43.270 prisudzujeme jej hodnotu nula. 00:02:43.270 --> 00:02:44.820 Hovorime tomu funkcia. Kedy sa 00:02:44.820 --> 00:02:48.220 tato funkcia rovna 0? 00:02:48.220 --> 00:02:49.390 kedy? 00:02:49.390 --> 00:02:51.720 Rovna sa nule v tychto bodoch, ano? 00:02:51.720 --> 00:02:55.360 Pretoze tu sa f(x) rovna 0. 00:02:55.360 --> 00:02:57.490 Ked sme toto vyriesili 00:02:57.490 --> 00:03:01.970 rozlozenim, prisli sme na to, ze hodnoty x, ktore tvorili f(x), 00:03:01.970 --> 00:03:04.160 sa rovnaju 0, co su tieto dva body. 00:03:04.160 --> 00:03:06.740 Teraz trocha terminologie - nazyvaju sa 00:03:06.740 --> 00:03:09.860 nulami, alebo aj korenmi f(x). 00:03:09.860 --> 00:03:12.470 Trocha si to zopakujme. 00:03:14.810 --> 00:03:23.700 Ak by som mal nieco ako f(x) sa rovna x na druhu plus 00:03:23.700 --> 00:03:29.550 4 krat x plus 4, a opytal by som sa ta, kde su nuly ci 00:03:29.550 --> 00:03:31.770 korene f(x)? 00:03:31.770 --> 00:03:33.970 To je to iste, ako opytat sa ta: kde f(x) 00:03:33.970 --> 00:03:36.300 pretina os x? 00:03:36.300 --> 00:03:38.210 Pretina ju, ked f(x) 00:03:38.210 --> 00:03:39.440 sa rovna 0, ano? 00:03:39.440 --> 00:03:42.120 Ak teda myslime graf, ktory som predtym nakreslil. 00:03:42.120 --> 00:03:45.720 Povedzme, ze f(x) sa rovna 0, potom mozeme 00:03:45.720 --> 00:03:51.860 povedat, ze 0 sa rovna x na druhu plus 4 krat x plus 4. 00:03:51.860 --> 00:03:53.940 Vieme, ze to mozeme rozlozit, teda x 00:03:53.940 --> 00:03:57.080 plus 2 krat x plus 2. 00:03:57.080 --> 00:04:07.090 Vieme, ze sa to rovna 0, ak sa x rovna minus 2. 00:04:07.090 --> 00:04:10.170 x sa rovna minus 2. 00:04:13.940 --> 00:04:18.270 No, toto je trocha preklep, takze x sa rovna minus 2. 00:04:18.270 --> 00:04:22.380 Tak teraz uz vieme, ako najdeme korene, ked sa urcita 00:04:22.380 --> 00:04:24.560 rovnica da lahko rozlozit. 00:04:24.560 --> 00:04:27.500 Ale skusme rovnicu, ktoru nie je v skutocnosti 00:04:27.500 --> 00:04:28.850 take lahke rozlozit. 00:04:28.850 --> 00:04:32.120 Priklad: mame f(x) sa rovna minus 10 krat x 00:04:39.750 --> 00:04:45.380 na druhu minus 9 krat x plus 1. 00:04:45.380 --> 00:04:47.580 Ked sa na to pozriem, aj keby som to vydelil 10, 00:04:47.580 --> 00:04:48.650 ostali by mi tu nejake zlomky. 00:04:48.650 --> 00:04:53.130 Je velmi tazke predstavit si rozlozenie tejto kvadratickej rovnice. 00:04:53.130 --> 00:04:54.860 Toto sa vlastne vola kvadraticka rovnica, alebo 00:04:54.860 --> 00:04:57.580 druhostupnovy polynomial. 00:04:57.580 --> 00:04:59.600 Skusime to vyriesit. 00:04:59.600 --> 00:05:02.420 Pretoze chceme zistit, kedy sa to rovna 0. 00:05:02.420 --> 00:05:07.130 Minus 10 krat x na druhu minus 9 krat x plus 1. 00:05:07.130 --> 00:05:09.090 Chceme zistit, ake hodnoty musi mat x, aby 00:05:09.090 --> 00:05:11.260 sa tato rovnica rovnala 0. 00:05:11.260 --> 00:05:13.730 A tu mozme pouzit pomocku nazvanu vzorec kvadratickej rovnice. 00:05:13.730 --> 00:05:15.625 Teraz vam dam jednu radu v matematike, 00:05:15.625 --> 00:05:18.030 ktoru je dobre si zapamatat. 00:05:18.030 --> 00:05:21.330 Korene kvadratickej rovnice sa vypocitaju podla daneho vzorca. 00:05:21.330 --> 00:05:24.810 Kvadraticka rovnica ma vo vseobecnosti takyto tvar: 00:05:24.810 --> 00:05:31.900 A krat x na druhu plus B krat x plus C sa rovna 0. 00:05:31.900 --> 00:05:35.790 V nasom priklade je A minus 10, 00:05:35.790 --> 00:05:39.940 B je minus 9, a C je 1. 00:05:39.940 --> 00:05:48.040 Vzorec je: korene x sa rovnaju minus B plus alebo minus 00:05:48.040 --> 00:05:58.060 druha odmocnina B na druhu minus 4 krat A krat C, 00:05:58.060 --> 00:06:00.230 vsetko to delene 2 krat A. 00:06:00.230 --> 00:06:02.843 Viem, ze to vyzera zlozito, ale cim viacej to budes pouzivat, 00:06:02.843 --> 00:06:04.400 uvidis, ze to v skutocnosti nie je az take zle. 00:06:04.400 --> 00:06:07.720 Je dobre si ten vzorec zapamatat. 00:06:07.720 --> 00:06:10.730 Aplikujme tento vzorec na nasu rovnicu, 00:06:10.730 --> 00:06:12.670 ktoru sme si napisali. 00:06:12.670 --> 00:06:15.260 Takze - pozri sa, A je iba koeficient 00:06:15.260 --> 00:06:18.610 clena x na druhu, ano? 00:06:18.610 --> 00:06:20.300 takze A je koeficient clena x na druhú. 00:06:20.300 --> 00:06:23.570 B je koeficient clena x. C je konštanta. 00:06:23.570 --> 00:06:25.100 Takze aplikujme tento vzorec na nasu rovnicu. 00:06:25.100 --> 00:06:26.250 Kolko je B? 00:06:26.250 --> 00:06:28.700 B je minus 9. 00:06:28.700 --> 00:06:29.970 Mozeme to vidiet tu. 00:06:29.970 --> 00:06:33.980 B je minus 9, A je minus 10. 00:06:33.980 --> 00:06:34.970 C je 1. 00:06:34.970 --> 00:06:36.090 Ano? 00:06:36.090 --> 00:06:42.350 Ak B je minus 9 - tak potom mame minus minus 9. 00:06:42.350 --> 00:06:49.260 Plus alebo mínus druhá odmocnina minus 9 na druhú. 00:06:49.260 --> 00:06:49.810 To je 81. 00:06:49.810 --> 00:06:53.140 Mínus 4 krát A. 00:06:56.940 --> 00:06:59.760 A je mínus 10. 00:06:59.760 --> 00:07:03.240 Mínus 10 krát C, ktore je 1. 00:07:03.240 --> 00:07:05.110 Viem, že je to chaoticke, ale dúfam, že to 00:07:05.110 --> 00:07:06.470 chapes. 00:07:06.470 --> 00:07:09.560 Všetko delene 2 krát A. 00:07:09.560 --> 00:07:14.050 A je mínus 10, takze 2 krát A je potom mínus 20. 00:07:14.050 --> 00:07:14.990 Tak si to zjednodušme. 00:07:14.990 --> 00:07:19.410 minus minus 9, to je kladne 9. 00:07:19.410 --> 00:07:26.460 Plus alebo mínus druhá odmocnina z 81. 00:07:26.460 --> 00:07:30.660 Máme minus 4 krat A, ktore je minus 10 . 00:07:30.660 --> 00:07:31.870 Tu je mínus 10. 00:07:31.870 --> 00:07:33.280 Viem, že je to veľmi komplikované, je mi to luto, 00:07:33.280 --> 00:07:34.380 krat C, teda krat 1. 00:07:34.380 --> 00:07:39.410 minus 4 krat minus 10 je 40, kladne 40. 00:07:39.410 --> 00:07:41.040 Kladne 40. 00:07:41.040 --> 00:07:46.070 To vsetko vydelime minus 20. . 00:07:46.070 --> 00:07:48.300 81 plus 40 je 121. 00:07:48.300 --> 00:07:52.330 9 plus alebo mínus druhá odmocnina 00:07:52.330 --> 00:07:58.290 zo 121 delene mínus 20. 00:07:58.290 --> 00:08:01.620 Druhá odmocnina zo 121 je 11. 00:08:01.620 --> 00:08:03.170 Pôjdem sem. 00:08:03.170 --> 00:08:06.184 Dúfam, že nestratís prehľad o tom, čo robím. 00:08:06.184 --> 00:08:13.720 9 plus alebo mínus 11, delene mínus 20. 00:08:13.720 --> 00:08:19.090 9 plus 11 delene mínus 20, to je 9 00:08:19.090 --> 00:08:22.540 plus 11 je 20, takže to je 20 delene mínus 20, 00:08:22.540 --> 00:08:23.730 co sa rovná minus 1 . 00:08:23.730 --> 00:08:24.900 Takže tu mame prvy koreň. 00:08:24.900 --> 00:08:28.260 To je 9 plus - pretože to je plus alebo mínus. 00:08:28.260 --> 00:08:33.790 A ten druhý koreň potom bude 9 mínus 11 delene minus 20, 00:08:33.790 --> 00:08:37.720 co sa rovná mínus 2 delene mínus 20, 00:08:37.720 --> 00:08:40.700 co sa rovná 1 lomene 10. 00:08:40.700 --> 00:08:42.690 Tak toto je dalsi koren. 00:08:42.690 --> 00:08:48.950 Ak by sme tuto rovnicu zobrazili na grafe, videli by sme, ze v 00:08:48.950 --> 00:08:52.640 bodoch minus 1 a 1/10 naozaj pretína os x. 00:08:52.640 --> 00:08:57.770 Alebo f ( x) sa rovna 0 v bodoch, kde x sa rovna 00:08:57.770 --> 00:09:01.690 minus 1 alebo x sa rovná 1/10. 00:09:01.690 --> 00:09:04.080 V časti 2 budu dalsie príklady, pretože si 00:09:04.080 --> 00:09:06.100 myslím, ze ak niečo, tak možno som ta 00:09:06.100 --> 00:09:08.120 tymto trocha doplietol. 00:09:08.120 --> 00:09:11.680 Uvidíme sa teda v časti 2 s dalsimi 00:09:11.680 --> 00:09:12.150 kvadratickymi rovnicami. 00:09:12.150 --> 00:09:14.083 ...