☰ Reja · Matematika

Algebraik kasrlar va ildizli ifodalarni soddalashtirish

8-hafta · blok 1.6 · 3 dars
3

Ildiz va daraja qatnashgan ifodalar

maktab darsligi: algebra 8–9-sinf, ayniy almashtirishlar
Maqsad: Ildizli va darajali ifodalarni soddalashtirish va qiymatini topish.
Qoida va g‘oya

Ildizli ifodalarda eng avval sqrt(a^2) = |a| ni eslang. Ildiz ostidagi ifoda to‘la kvadrat bo‘lsa, uni kvadratga aylantiring: sqrt(x^2 − 10x + 25) = sqrt((x − 5)^2) = |x − 5|. Modulni ochishda x ning oralig‘iga qarab ishorani aniqlang: x < 5 bo‘lsa |x − 5| = 5 − x. Maxrajda ildiz bo‘lsa, qo‘shma ifodaga ko‘paytiring. Ifodani avval soddalashtiring, keyin qiymatni qo‘ying. Ko‘p masalalarda yig‘indi yoki ayirma simmetrik bo‘lib, ildizlar o‘zaro yo‘qoladi.

Namunaviy masalalar
x = −5 da sqrt(4x^2) ning qiymatini toping.
Yechim: sqrt(4x^2) = 2|x| = 2 · 5 = 10.
Javob: 10
2 < x < 5 bo‘lganda sqrt(x^2 − 10x + 25) + |x − 2| ni soddalashtiring.
Yechim: sqrt((x − 5)^2) = |x − 5| = 5 − x. |x − 2| = x − 2. Yig‘indi 5 − x + x − 2 = 3.
Javob: 3
(sqrt(5) + 1) / (sqrt(5) − 1) ning maxrajini irratsionallikdan qutqaring.
Yechim: (sqrt(5) + 1)^2 / (5 − 1) = (6 + 2 sqrt(5)) / 4 = (3 + sqrt(5)) / 2.
Javob: (3 + sqrt(5)) / 2
Tipik xatolar
  • sqrt(x^2) = x deb yozish: x manfiy bo‘lsa natija −x.
  • Modulni ochishda oraliqni hisobga olmaslik.
  • Maxrajni ayirmaga ko‘paytirishda surat va maxrajning ikkalasini ko‘paytirmaslik.
Mashq
1
x = −4 da sqrt(9x^2) ning qiymatini toping.
2
(sqrt(7) + sqrt(3)) (sqrt(7) − sqrt(3)) ni hisoblang.
3
x < 1 bo‘lganda sqrt(x^2 − 2x + 1) ni soddalashtiring.
4
1/(sqrt(2) − 1) + 1/(sqrt(2) + 1) ni hisoblang.