3w^2+6=-11w eching

$$3 w ^ { 2 } + 6 = - 11 w$$

w uchun yechish

$$w=-3$$

$$w=-\frac{2}{3}\approx -0,666666667$$

Guruhlash orqali faktorlash qadamlari

11w ni ikki tarafga qo’shing.

$$3w^{2}+6+11w=0$$

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$3w^{2}+11w+6=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 3w^{2}+aw+bw+6 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=11 ab=3\times 6=18$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 18-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,18 2,9 3,6$$

Har bir juftlik yigʻindisini hisoblang.

$$1+18=19 2+9=11 3+6=9$$

Yechim – 11 yigʻindisini beruvchi juftlik.

$$a=2 b=9$$

3w^{2}+11w+6 ni \left(3w^{2}+2w\right)+\left(9w+6\right) sifatida qaytadan yozish.

$$\left(3w^{2}+2w\right)+\left(9w+6\right)$$

Birinchi guruhda w ni va ikkinchi guruhda 3 ni faktordan chiqaring.

$$w\left(3w+2\right)+3\left(3w+2\right)$$

Distributiv funktsiyasidan foydalangan holda 3w+2 umumiy terminini chiqaring.

$$\left(3w+2\right)\left(w+3\right)$$

Tenglamani yechish uchun 3w+2=0 va w+3=0 ni yeching.

$$w=-\frac{2}{3} w=-3$$

Kvadrat tenglamalaridan foydalanish qoidalari

11w ni ikki tarafga qo’shing.

$$3w^{2}+6+11w=0$$

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.

$$3w^{2}+11w+6=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, 11 ni b va 6 ni c bilan almashtiring.

$$w=\frac{-11±\sqrt{11^{2}-4\times 3\times 6}}{2\times 3}$$

11 kvadratini chiqarish.

$$w=\frac{-11±\sqrt{121-4\times 3\times 6}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$w=\frac{-11±\sqrt{121-12\times 6}}{2\times 3}$$

-12 ni 6 marotabaga ko'paytirish.

$$w=\frac{-11±\sqrt{121-72}}{2\times 3}$$

121 ni -72 ga qo'shish.

$$w=\frac{-11±\sqrt{49}}{2\times 3}$$

49 ning kvadrat ildizini chiqarish.

$$w=\frac{-11±7}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$w=\frac{-11±7}{6}$$

w=\frac{-11±7}{6} tenglamasini yeching, bunda ± musbat. -11 ni 7 ga qo'shish.

$$w=-\frac{4}{6}$$

\frac{-4}{6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$w=-\frac{2}{3}$$

w=\frac{-11±7}{6} tenglamasini yeching, bunda ± manfiy. -11 dan 7 ni ayirish.

$$w=-\frac{18}{6}$$

-18 ni 6 ga bo'lish.

$$w=-3$$

Tenglama yechildi.

$$w=-\frac{2}{3} w=-3$$

Kvadratni yakunlash bo'yicha qadamlar

11w ni ikki tarafga qo’shing.

$$3w^{2}+6+11w=0$$

Ikkala tarafdan 6 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$3w^{2}+11w=-6$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3w^{2}+11w}{3}=-\frac{6}{3}$$

3 ga bo'lish 3 ga ko'paytirishni bekor qiladi.

$$w^{2}+\frac{11}{3}w=-\frac{6}{3}$$

-6 ni 3 ga bo'lish.

$$w^{2}+\frac{11}{3}w=-2$$

\frac{11}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{11}{6} olish uchun. Keyin, \frac{11}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$w^{2}+\frac{11}{3}w+\left(\frac{11}{6}\right)^{2}=-2+\left(\frac{11}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{11}{6} kvadratini chiqarish.

$$w^{2}+\frac{11}{3}w+\frac{121}{36}=-2+\frac{121}{36}$$

-2 ni \frac{121}{36} ga qo'shish.

$$w^{2}+\frac{11}{3}w+\frac{121}{36}=\frac{49}{36}$$

w^{2}+\frac{11}{3}w+\frac{121}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(w+\frac{11}{6}\right)^{2}=\frac{49}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(w+\frac{11}{6}\right)^{2}}=\sqrt{\frac{49}{36}}$$

Qisqartirish.

$$w+\frac{11}{6}=\frac{7}{6} w+\frac{11}{6}=-\frac{7}{6}$$

Tenglamaning ikkala tarafidan \frac{11}{6} ni ayirish.

$$w=-\frac{2}{3} w=-3$$