1. à) õ2 – 10õ + 21 = (õ – 3) (õ – 7);
õ2 – 10õ + 21 = 0;
õ1 = 3, õ2 = 7.
á) 5ó2 + 9ó – 2 = 5 (ó – ) (ó + 2) = (5ó – 1) (ó + 2);
5ó2 + 9ó – 2 = 0;
D = 81 + 40 = 121;
ó1, 2 = ;
ó1 = , ó2 = –2.
2. ó = õ2 – 4õ – 5 – êâàäðàòè÷íàÿ ôóíêöèÿ, ãðàôèêîì ÿâëÿåòñÿ ïàðàáîëà.
Òàê êàê à > 0, òî âåòâè íàïðàâëåíû ââåðõ. Íàéäåì êîîðäèíàòû (ò; ï) âåðøèíû ïàðàáîëû:
m = = 2; ï = 4 – 8 – 5 = –9;
À (2; –9) – âåðøèíà ïàðàáîëû.
õ | –1 | –2 | ||
ó | –8 | –5 |
à) ó ≈ –6; á) õ ≈ –1,5; 5,3; â) ó = 0 ïðè õ = –1 è õ = 5; ã) ó > 0 ïðè õ (–∞; –1) (5; +∞); ó < 0 ïðè õ (–1; 5); ä) (–∞; 2]. |
3. à) (–1,7)5 > (–2,1)5; â) 4,79 > ;
á) > ; ã) 5,712 < (–6,3)12.
4. à) ;
á) ;
â) .
5. ;
4ñ2 + 7ñ – 2 = 0;
D = 49 + 32 = 81;
ñ1, 2 = ;
ñ1 = , ñ2 = –2.
6. –õ2 + 4õ + 3.
1-é ñ ï î ñ î á.
Âûäåëèì êâàäðàò äâó÷ëåíà èç êâàäðàòíîãî òðåõ÷ëåíà:
–õ2 + 4õ + 3 = –(õ2 – 4õ – 3) = –(õ2 –2 · 2 · õ + 4 – 4 –3) = –((õ – 2)2 – 7) =
= –(õ – 2)2 + 7.
Ýòî âûðàæåíèå ïðèíèìàåò íàèáîëüøåå çíà÷åíèå ïðè õ = 2, è îíî ðàâíî 7.
2-é ñ ï î ñ î á.
ó = –õ2 + 4õ + 3 – êâàäðàòè÷íàÿ ôóíêöèÿ, ãðàôèêîì ÿâëÿåòñÿ ïàðàáîëà, âåòâè êîòîðîé íàïðàâëåíû âíèç. Íàèáîëüøåå çíà÷åíèå êâàäðàòíîãî òðåõ÷ëåíà –õ2 + 4õ + 3 – ýòî îðäèíàòà âåðøèíû ýòîé ïàðàáîëû:
ò = = 2; ï = –4 + 8 + 3 = 7;
7 – íàèáîëüøåå çíà÷åíèå êâàäðàòíîãî òðåõ÷ëåíà –õ2 + 4õ + 3.