left(3x+1right)^2=-2x eching

$${ \left(3x+1 \right) }^{ 2 } =-2x$$

x uchun yechish

$$x=\frac{\sqrt{7}-4}{9}\approx -0,150472077$$

$$x=\frac{-\sqrt{7}-4}{9}\approx -0,738416812$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(3x+1\right)^{2} kengaytirilishi uchun ishlating.

$$9x^{2}+6x+1=-2x$$

2x ni ikki tarafga qo’shing.

$$9x^{2}+6x+1+2x=0$$

8x ni olish uchun 6x va 2x ni birlashtirish.

$$9x^{2}+8x+1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 9 ni a, 8 ni b va 1 ni c bilan almashtiring.

$$x=\frac{-8±\sqrt{8^{2}-4\times 9}}{2\times 9}$$

8 kvadratini chiqarish.

$$x=\frac{-8±\sqrt{64-4\times 9}}{2\times 9}$$

-4 ni 9 marotabaga ko'paytirish.

$$x=\frac{-8±\sqrt{64-36}}{2\times 9}$$

64 ni -36 ga qo'shish.

$$x=\frac{-8±\sqrt{28}}{2\times 9}$$

28 ning kvadrat ildizini chiqarish.

$$x=\frac{-8±2\sqrt{7}}{2\times 9}$$

2 ni 9 marotabaga ko'paytirish.

$$x=\frac{-8±2\sqrt{7}}{18}$$

x=\frac{-8±2\sqrt{7}}{18} tenglamasini yeching, bunda ± musbat. -8 ni 2\sqrt{7} ga qo'shish.

$$x=\frac{2\sqrt{7}-8}{18}$$

-8+2\sqrt{7} ni 18 ga bo'lish.

$$x=\frac{\sqrt{7}-4}{9}$$

x=\frac{-8±2\sqrt{7}}{18} tenglamasini yeching, bunda ± manfiy. -8 dan 2\sqrt{7} ni ayirish.

$$x=\frac{-2\sqrt{7}-8}{18}$$

-8-2\sqrt{7} ni 18 ga bo'lish.

$$x=\frac{-\sqrt{7}-4}{9}$$

Tenglama yechildi.

$$x=\frac{\sqrt{7}-4}{9} x=\frac{-\sqrt{7}-4}{9}$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(3x+1\right)^{2} kengaytirilishi uchun ishlating.

$$9x^{2}+6x+1=-2x$$

2x ni ikki tarafga qo’shing.

$$9x^{2}+6x+1+2x=0$$

8x ni olish uchun 6x va 2x ni birlashtirish.

$$9x^{2}+8x+1=0$$

Ikkala tarafdan 1 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$9x^{2}+8x=-1$$

Ikki tarafini 9 ga bo‘ling.

$$\frac{9x^{2}+8x}{9}=-\frac{1}{9}$$

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

$$x^{2}+\frac{8}{9}x=-\frac{1}{9}$$

\frac{8}{9} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{4}{9} olish uchun. Keyin, \frac{4}{9} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{8}{9}x+\left(\frac{4}{9}\right)^{2}=-\frac{1}{9}+\left(\frac{4}{9}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{4}{9} kvadratini chiqarish.

$$x^{2}+\frac{8}{9}x+\frac{16}{81}=-\frac{1}{9}+\frac{16}{81}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{9} ni \frac{16}{81} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{8}{9}x+\frac{16}{81}=\frac{7}{81}$$

x^{2}+\frac{8}{9}x+\frac{16}{81} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{4}{9}\right)^{2}=\frac{7}{81}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{4}{9}\right)^{2}}=\sqrt{\frac{7}{81}}$$

Qisqartirish.

$$x+\frac{4}{9}=\frac{\sqrt{7}}{9} x+\frac{4}{9}=-\frac{\sqrt{7}}{9}$$

Tenglamaning ikkala tarafidan \frac{4}{9} ni ayirish.

$$x=\frac{\sqrt{7}-4}{9} x=\frac{-\sqrt{7}-4}{9}$$