3m^2=-16m-21 eching

$$3 m ^ { 2 } = - 16 m - 21$$

m uchun yechish

$$m = -\frac{7}{3} = -2\frac{1}{3} \approx -2,333333333$$

$$m=-3$$

Guruhlash orqali faktorlash qadamlari

16m ni ikki tarafga qo’shing.

$$3m^{2}+16m=-21$$

21 ni ikki tarafga qo’shing.

$$3m^{2}+16m+21=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 3m^{2}+am+bm+21 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=16 ab=3\times 21=63$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 63-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,63 3,21 7,9$$

Har bir juftlik yigʻindisini hisoblang.

$$1+63=64 3+21=24 7+9=16$$

Yechim – 16 yigʻindisini beruvchi juftlik.

$$a=7 b=9$$

3m^{2}+16m+21 ni \left(3m^{2}+7m\right)+\left(9m+21\right) sifatida qaytadan yozish.

$$\left(3m^{2}+7m\right)+\left(9m+21\right)$$

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

$$m\left(3m+7\right)+3\left(3m+7\right)$$

Distributiv funktsiyasidan foydalangan holda 3m+7 umumiy terminini chiqaring.

$$\left(3m+7\right)\left(m+3\right)$$

Tenglamani yechish uchun 3m+7=0 va m+3=0 ni yeching.

$$m=-\frac{7}{3} m=-3$$

Kvadrat tenglamalaridan foydalanish qoidalari

16m ni ikki tarafga qo’shing.

$$3m^{2}+16m=-21$$

21 ni ikki tarafga qo’shing.

$$3m^{2}+16m+21=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, 16 ni b va 21 ni c bilan almashtiring.

$$m=\frac{-16±\sqrt{16^{2}-4\times 3\times 21}}{2\times 3}$$

16 kvadratini chiqarish.

$$m=\frac{-16±\sqrt{256-4\times 3\times 21}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$m=\frac{-16±\sqrt{256-12\times 21}}{2\times 3}$$

-12 ni 21 marotabaga ko'paytirish.

$$m=\frac{-16±\sqrt{256-252}}{2\times 3}$$

256 ni -252 ga qo'shish.

$$m=\frac{-16±\sqrt{4}}{2\times 3}$$

4 ning kvadrat ildizini chiqarish.

$$m=\frac{-16±2}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$m=\frac{-16±2}{6}$$

m=\frac{-16±2}{6} tenglamasini yeching, bunda ± musbat. -16 ni 2 ga qo'shish.

$$m=-\frac{14}{6}$$

\frac{-14}{6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$m=-\frac{7}{3}$$

m=\frac{-16±2}{6} tenglamasini yeching, bunda ± manfiy. -16 dan 2 ni ayirish.

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

-18 ni 6 ga bo'lish.

$$m=-3$$

Tenglama yechildi.

$$m=-\frac{7}{3} m=-3$$

Kvadratni yakunlash bo'yicha qadamlar

16m ni ikki tarafga qo’shing.

$$3m^{2}+16m=-21$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3m^{2}+16m}{3}=-\frac{21}{3}$$

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

$$m^{2}+\frac{16}{3}m=-\frac{21}{3}$$

-21 ni 3 ga bo'lish.

$$m^{2}+\frac{16}{3}m=-7$$

\frac{16}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{8}{3} olish uchun. Keyin, \frac{8}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$m^{2}+\frac{16}{3}m+\left(\frac{8}{3}\right)^{2}=-7+\left(\frac{8}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{8}{3} kvadratini chiqarish.

$$m^{2}+\frac{16}{3}m+\frac{64}{9}=-7+\frac{64}{9}$$

-7 ni \frac{64}{9} ga qo'shish.

$$m^{2}+\frac{16}{3}m+\frac{64}{9}=\frac{1}{9}$$

m^{2}+\frac{16}{3}m+\frac{64}{9} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(m+\frac{8}{3}\right)^{2}=\frac{1}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(m+\frac{8}{3}\right)^{2}}=\sqrt{\frac{1}{9}}$$

Qisqartirish.

$$m+\frac{8}{3}=\frac{1}{3} m+\frac{8}{3}=-\frac{1}{3}$$

Tenglamaning ikkala tarafidan \frac{8}{3} ni ayirish.

$$m=-\frac{7}{3} m=-3$$