9x^2-19x-24=0 eching

$$9 x ^ { 2 } - 19 x - 24 = 0$$

x uchun yechish

$$x=-\frac{8}{9}\approx -0,888888889$$

$$x=3$$

Guruhlash orqali faktorlash qadamlari

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 9x^{2}+ax+bx-24 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-19 ab=9\left(-24\right)=-216$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -216-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-216 2,-108 3,-72 4,-54 6,-36 8,-27 9,-24 12,-18$$

Har bir juftlik yigʻindisini hisoblang.

$$1-216=-215 2-108=-106 3-72=-69 4-54=-50 6-36=-30 8-27=-19 9-24=-15 12-18=-6$$

Yechim – -19 yigʻindisini beruvchi juftlik.

$$a=-27 b=8$$

9x^{2}-19x-24 ni \left(9x^{2}-27x\right)+\left(8x-24\right) sifatida qaytadan yozish.

$$\left(9x^{2}-27x\right)+\left(8x-24\right)$$

Birinchi guruhda 9x ni va ikkinchi guruhda 8 ni faktordan chiqaring.

$$9x\left(x-3\right)+8\left(x-3\right)$$

Distributiv funktsiyasidan foydalangan holda x-3 umumiy terminini chiqaring.

$$\left(x-3\right)\left(9x+8\right)$$

Tenglamani yechish uchun x-3=0 va 9x+8=0 ni yeching.

$$x=3 x=-\frac{8}{9}$$

Kvadrat tenglamalaridan foydalanish qoidalari

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$9x^{2}-19x-24=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 9 ni a, -19 ni b va -24 ni c bilan almashtiring.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 9\left(-24\right)}}{2\times 9}$$

-19 kvadratini chiqarish.

$$x=\frac{-\left(-19\right)±\sqrt{361-4\times 9\left(-24\right)}}{2\times 9}$$

-4 ni 9 marotabaga ko'paytirish.

$$x=\frac{-\left(-19\right)±\sqrt{361-36\left(-24\right)}}{2\times 9}$$

-36 ni -24 marotabaga ko'paytirish.

$$x=\frac{-\left(-19\right)±\sqrt{361+864}}{2\times 9}$$

361 ni 864 ga qo'shish.

$$x=\frac{-\left(-19\right)±\sqrt{1225}}{2\times 9}$$

1225 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-19\right)±35}{2\times 9}$$

-19 ning teskarisi 19 ga teng.

$$x=\frac{19±35}{2\times 9}$$

2 ni 9 marotabaga ko'paytirish.

$$x=\frac{19±35}{18}$$

x=\frac{19±35}{18} tenglamasini yeching, bunda ± musbat. 19 ni 35 ga qo'shish.

$$x=\frac{54}{18}$$

54 ni 18 ga bo'lish.

$$x=3$$

x=\frac{19±35}{18} tenglamasini yeching, bunda ± manfiy. 19 dan 35 ni ayirish.

$$x=-\frac{16}{18}$$

\frac{-16}{18} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{8}{9}$$

Tenglama yechildi.

$$x=3 x=-\frac{8}{9}$$

Kvadratni yakunlash bo'yicha qadamlar

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$9x^{2}-19x-24=0$$

24 ni tenglamaning ikkala tarafiga qo'shish.

$$9x^{2}-19x-24-\left(-24\right)=-\left(-24\right)$$

O‘zidan -24 ayirilsa 0 qoladi.

$$9x^{2}-19x=-\left(-24\right)$$

0 dan -24 ni ayirish.

$$9x^{2}-19x=24$$

Ikki tarafini 9 ga bo‘ling.

$$\frac{9x^{2}-19x}{9}=\frac{24}{9}$$

9 ga bo'lish 9 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{19}{9}x=\frac{24}{9}$$

\frac{24}{9} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{19}{9}x=\frac{8}{3}$$

-\frac{19}{9} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{19}{18} olish uchun. Keyin, -\frac{19}{18} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{19}{9}x+\left(-\frac{19}{18}\right)^{2}=\frac{8}{3}+\left(-\frac{19}{18}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{19}{18} kvadratini chiqarish.

$$x^{2}-\frac{19}{9}x+\frac{361}{324}=\frac{8}{3}+\frac{361}{324}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{8}{3} ni \frac{361}{324} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{19}{9}x+\frac{361}{324}=\frac{1225}{324}$$

x^{2}-\frac{19}{9}x+\frac{361}{324} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{19}{18}\right)^{2}=\frac{1225}{324}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{19}{18}\right)^{2}}=\sqrt{\frac{1225}{324}}$$

Qisqartirish.

$$x-\frac{19}{18}=\frac{35}{18} x-\frac{19}{18}=-\frac{35}{18}$$

\frac{19}{18} ni tenglamaning ikkala tarafiga qo'shish.

$$x=3 x=-\frac{8}{9}$$