8v^2-21v-3=6 eching

$$8 v ^ { 2 } - 21 v - 3 = 6$$

v uchun yechish

$$v=-\frac{3}{8}=-0,375$$

$$v=3$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan 6 ni ayirish.

$$8v^{2}-21v-3-6=0$$

-9 olish uchun -3 dan 6 ni ayirish.

$$8v^{2}-21v-9=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 8v^{2}+av+bv-9 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-21 ab=8\left(-9\right)=-72$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -72-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-72 2,-36 3,-24 4,-18 6,-12 8,-9$$

Har bir juftlik yigʻindisini hisoblang.

$$1-72=-71 2-36=-34 3-24=-21 4-18=-14 6-12=-6 8-9=-1$$

Yechim – -21 yigʻindisini beruvchi juftlik.

$$a=-24 b=3$$

8v^{2}-21v-9 ni \left(8v^{2}-24v\right)+\left(3v-9\right) sifatida qaytadan yozish.

$$\left(8v^{2}-24v\right)+\left(3v-9\right)$$

Birinchi guruhda 8v ni va ikkinchi guruhda 3 ni faktordan chiqaring.

$$8v\left(v-3\right)+3\left(v-3\right)$$

Distributiv funktsiyasidan foydalangan holda v-3 umumiy terminini chiqaring.

$$\left(v-3\right)\left(8v+3\right)$$

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

$$v=3 v=-\frac{3}{8}$$

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.

$$8v^{2}-21v-3=6$$

Tenglamaning ikkala tarafidan 6 ni ayirish.

$$8v^{2}-21v-3-6=6-6$$

O‘zidan 6 ayirilsa 0 qoladi.

$$8v^{2}-21v-3-6=0$$

-3 dan 6 ni ayirish.

$$8v^{2}-21v-9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 8 ni a, -21 ni b va -9 ni c bilan almashtiring.

$$v=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 8\left(-9\right)}}{2\times 8}$$

-21 kvadratini chiqarish.

$$v=\frac{-\left(-21\right)±\sqrt{441-4\times 8\left(-9\right)}}{2\times 8}$$

-4 ni 8 marotabaga ko'paytirish.

$$v=\frac{-\left(-21\right)±\sqrt{441-32\left(-9\right)}}{2\times 8}$$

-32 ni -9 marotabaga ko'paytirish.

$$v=\frac{-\left(-21\right)±\sqrt{441+288}}{2\times 8}$$

441 ni 288 ga qo'shish.

$$v=\frac{-\left(-21\right)±\sqrt{729}}{2\times 8}$$

729 ning kvadrat ildizini chiqarish.

$$v=\frac{-\left(-21\right)±27}{2\times 8}$$

-21 ning teskarisi 21 ga teng.

$$v=\frac{21±27}{2\times 8}$$

2 ni 8 marotabaga ko'paytirish.

$$v=\frac{21±27}{16}$$

v=\frac{21±27}{16} tenglamasini yeching, bunda ± musbat. 21 ni 27 ga qo'shish.

$$v=\frac{48}{16}$$

48 ni 16 ga bo'lish.

$$v=3$$

v=\frac{21±27}{16} tenglamasini yeching, bunda ± manfiy. 21 dan 27 ni ayirish.

$$v=-\frac{6}{16}$$

\frac{-6}{16} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$v=-\frac{3}{8}$$

Tenglama yechildi.

$$v=3 v=-\frac{3}{8}$$

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.

$$8v^{2}-21v-3=6$$

3 ni tenglamaning ikkala tarafiga qo'shish.

$$8v^{2}-21v-3-\left(-3\right)=6-\left(-3\right)$$

O‘zidan -3 ayirilsa 0 qoladi.

$$8v^{2}-21v=6-\left(-3\right)$$

6 dan -3 ni ayirish.

$$8v^{2}-21v=9$$

Ikki tarafini 8 ga bo‘ling.

$$\frac{8v^{2}-21v}{8}=\frac{9}{8}$$

8 ga bo'lish 8 ga ko'paytirishni bekor qiladi.

$$v^{2}-\frac{21}{8}v=\frac{9}{8}$$

-\frac{21}{8} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{21}{16} olish uchun. Keyin, -\frac{21}{16} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$v^{2}-\frac{21}{8}v+\left(-\frac{21}{16}\right)^{2}=\frac{9}{8}+\left(-\frac{21}{16}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{21}{16} kvadratini chiqarish.

$$v^{2}-\frac{21}{8}v+\frac{441}{256}=\frac{9}{8}+\frac{441}{256}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{9}{8} ni \frac{441}{256} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$v^{2}-\frac{21}{8}v+\frac{441}{256}=\frac{729}{256}$$

v^{2}-\frac{21}{8}v+\frac{441}{256} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(v-\frac{21}{16}\right)^{2}=\frac{729}{256}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(v-\frac{21}{16}\right)^{2}}=\sqrt{\frac{729}{256}}$$

Qisqartirish.

$$v-\frac{21}{16}=\frac{27}{16} v-\frac{21}{16}=-\frac{27}{16}$$

\frac{21}{16} ni tenglamaning ikkala tarafiga qo'shish.

$$v=3 v=-\frac{3}{8}$$