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Irratsional tenglamalar
maktab darsligi: matematika 10-sinf, irratsional tenglamalar
Maqsad: Irratsional tenglamani kvadratga ko‘tarib yechish va begona ildizlarni tekshiruv bilan tashlash.
Qoida va g‘oya
Irratsional tenglamada noma’lum ildiz ostida bo‘ladi. sqrt(f(x)) = g(x) ni yechish: ikki tomonni kvadratga ko‘taramiz, f(x) = g(x)^2 ni yechamiz. Kvadratga ko‘tarish begona ildiz keltirishi mumkin, shuning uchun har doim tekshiring: g(x) ≥ 0 bo‘lishi va ildiz osti ≥ 0 bo‘lishi kerak. Masalan, sqrt(x + 7) = x + 1 dan x + 7 = x^2 + 2x + 1, x^2 + x − 6 = 0, x = 2 yoki x = −3. x = −3 da chap tomon sqrt(4) = 2, o‘ng tomon −2: begona ildiz. Javob x = 2. Ildiz ikki marta qatnashsa, avval bittasini yolg‘iz qoldiring.
Namunaviy masalalar
sqrt(3x + 1) = 4 ni yeching.
Yechim: 3x + 1 = 16, x = 5. Tekshirish: sqrt(16) = 4.
Javob: 5
Yechim: 3x + 1 = 16, x = 5. Tekshirish: sqrt(16) = 4.
Javob: 5
sqrt(x + 7) = x + 1 ni yeching.
Yechim: x^2 + x − 6 = 0, x = 2 yoki −3. x = −3 begona, chunki x + 1 = −2 < 0.
Javob: 2
Yechim: x^2 + x − 6 = 0, x = 2 yoki −3. x = −3 begona, chunki x + 1 = −2 < 0.
Javob: 2
sqrt(2x + 3) = x ni yeching.
Yechim: 2x + 3 = x^2, x^2 − 2x − 3 = 0, x = 3 yoki −1. x = −1 begona (x ≥ 0 kerak).
Javob: 3
Yechim: 2x + 3 = x^2, x^2 − 2x − 3 = 0, x = 3 yoki −1. x = −1 begona (x ≥ 0 kerak).
Javob: 3
Tipik xatolar
- Kvadratga ko‘tarib, begona ildizni tekshirmaslik.
- O‘ng tomon manfiy bo‘lganda ham ildiz topishga urinish: sqrt(...) = −2 yechimga ega emas.
- Kvadratga ko‘tarishda (x + 1)^2 ni x^2 + 1 deb yozish.
Mashq
1
sqrt(x − 2) = 3 ni yeching.
11
2
sqrt(x) = −4 ni yeching.
yechim yo‘q
3
sqrt(2x + 1) = x − 1 ni yeching.
4
4
sqrt(x + 6) = x ni yeching.
3