3m^2+4m+1=5/9 eching

$$3 m ^ { 2 } + 4 m + 1 = \frac { 5 } { 9 }$$

m uchun yechish

$$m=\frac{2\sqrt{6}}{9}-\frac{2}{3}\approx -0,122335613$$

$$m=-\frac{2\sqrt{6}}{9}-\frac{2}{3}\approx -1,210997721$$

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.

$$3m^{2}+4m+1=\frac{5}{9}$$

Tenglamaning ikkala tarafidan \frac{5}{9} ni ayirish.

$$3m^{2}+4m+1-\frac{5}{9}=\frac{5}{9}-\frac{5}{9}$$

O‘zidan \frac{5}{9} ayirilsa 0 qoladi.

$$3m^{2}+4m+1-\frac{5}{9}=0$$

1 dan \frac{5}{9} ni ayirish.

$$3m^{2}+4m+\frac{4}{9}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, 4 ni b va \frac{4}{9} ni c bilan almashtiring.

$$m=\frac{-4±\sqrt{4^{2}-4\times 3\times \frac{4}{9}}}{2\times 3}$$

4 kvadratini chiqarish.

$$m=\frac{-4±\sqrt{16-4\times 3\times \frac{4}{9}}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$m=\frac{-4±\sqrt{16-12\times \frac{4}{9}}}{2\times 3}$$

-12 ni \frac{4}{9} marotabaga ko'paytirish.

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

16 ni -\frac{16}{3} ga qo'shish.

$$m=\frac{-4±\sqrt{\frac{32}{3}}}{2\times 3}$$

\frac{32}{3} ning kvadrat ildizini chiqarish.

$$m=\frac{-4±\frac{4\sqrt{6}}{3}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$m=\frac{-4±\frac{4\sqrt{6}}{3}}{6}$$

m=\frac{-4±\frac{4\sqrt{6}}{3}}{6} tenglamasini yeching, bunda ± musbat. -4 ni \frac{4\sqrt{6}}{3} ga qo'shish.

$$m=\frac{\frac{4\sqrt{6}}{3}-4}{6}$$

-4+\frac{4\sqrt{6}}{3} ni 6 ga bo'lish.

$$m=\frac{2\sqrt{6}}{9}-\frac{2}{3}$$

m=\frac{-4±\frac{4\sqrt{6}}{3}}{6} tenglamasini yeching, bunda ± manfiy. -4 dan \frac{4\sqrt{6}}{3} ni ayirish.

$$m=\frac{-\frac{4\sqrt{6}}{3}-4}{6}$$

-4-\frac{4\sqrt{6}}{3} ni 6 ga bo'lish.

$$m=-\frac{2\sqrt{6}}{9}-\frac{2}{3}$$

Tenglama yechildi.

$$m=\frac{2\sqrt{6}}{9}-\frac{2}{3} m=-\frac{2\sqrt{6}}{9}-\frac{2}{3}$$

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.

$$3m^{2}+4m+1=\frac{5}{9}$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$3m^{2}+4m+1-1=\frac{5}{9}-1$$

O‘zidan 1 ayirilsa 0 qoladi.

$$3m^{2}+4m=\frac{5}{9}-1$$

\frac{5}{9} dan 1 ni ayirish.

$$3m^{2}+4m=-\frac{4}{9}$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3m^{2}+4m}{3}=-\frac{\frac{4}{9}}{3}$$

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

$$m^{2}+\frac{4}{3}m=-\frac{\frac{4}{9}}{3}$$

-\frac{4}{9} ni 3 ga bo'lish.

$$m^{2}+\frac{4}{3}m=-\frac{4}{27}$$

\frac{4}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{2}{3} olish uchun. Keyin, \frac{2}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$m^{2}+\frac{4}{3}m+\left(\frac{2}{3}\right)^{2}=-\frac{4}{27}+\left(\frac{2}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{2}{3} kvadratini chiqarish.

$$m^{2}+\frac{4}{3}m+\frac{4}{9}=-\frac{4}{27}+\frac{4}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{4}{27} ni \frac{4}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$m^{2}+\frac{4}{3}m+\frac{4}{9}=\frac{8}{27}$$

m^{2}+\frac{4}{3}m+\frac{4}{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.

$$\left(m+\frac{2}{3}\right)^{2}=\frac{8}{27}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(m+\frac{2}{3}\right)^{2}}=\sqrt{\frac{8}{27}}$$

Qisqartirish.

$$m+\frac{2}{3}=\frac{2\sqrt{6}}{9} m+\frac{2}{3}=-\frac{2\sqrt{6}}{9}$$

Tenglamaning ikkala tarafidan \frac{2}{3} ni ayirish.

$$m=\frac{2\sqrt{6}}{9}-\frac{2}{3} m=-\frac{2\sqrt{6}}{9}-\frac{2}{3}$$