-(-36)/left(1+3xright)^2=1/3 eching

$$- \frac{ (-36) }{ { \left(1+3x \right) }^{ 2 } } = \frac{ 1 }{ 3 }$$

x uchun yechish

$$x=-2\sqrt{3}-\frac{1}{3}\approx -3,797434948$$

$$x=2\sqrt{3}-\frac{1}{3}\approx 3,130768282$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -\frac{1}{3} teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 3\left(3x+1\right)^{2} ga, \left(1+3x\right)^{2},3 ning eng kichik karralisiga ko‘paytiring.

$$-3\left(-36\right)=\left(3x+1\right)^{2}$$

108 hosil qilish uchun -3 va -36 ni ko'paytirish.

$$108=\left(3x+1\right)^{2}$$

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

$$108=9x^{2}+6x+1$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$9x^{2}+6x+1=108$$

Ikkala tarafdan 108 ni ayirish.

$$9x^{2}+6x+1-108=0$$

-107 olish uchun 1 dan 108 ni ayirish.

$$9x^{2}+6x-107=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 9 ni a, 6 ni b va -107 ni c bilan almashtiring.

$$x=\frac{-6±\sqrt{6^{2}-4\times 9\left(-107\right)}}{2\times 9}$$

6 kvadratini chiqarish.

$$x=\frac{-6±\sqrt{36-4\times 9\left(-107\right)}}{2\times 9}$$

-4 ni 9 marotabaga ko'paytirish.

$$x=\frac{-6±\sqrt{36-36\left(-107\right)}}{2\times 9}$$

-36 ni -107 marotabaga ko'paytirish.

$$x=\frac{-6±\sqrt{36+3852}}{2\times 9}$$

36 ni 3852 ga qo'shish.

$$x=\frac{-6±\sqrt{3888}}{2\times 9}$$

3888 ning kvadrat ildizini chiqarish.

$$x=\frac{-6±36\sqrt{3}}{2\times 9}$$

2 ni 9 marotabaga ko'paytirish.

$$x=\frac{-6±36\sqrt{3}}{18}$$

x=\frac{-6±36\sqrt{3}}{18} tenglamasini yeching, bunda ± musbat. -6 ni 36\sqrt{3} ga qo'shish.

$$x=\frac{36\sqrt{3}-6}{18}$$

-6+36\sqrt{3} ni 18 ga bo'lish.

$$x=2\sqrt{3}-\frac{1}{3}$$

x=\frac{-6±36\sqrt{3}}{18} tenglamasini yeching, bunda ± manfiy. -6 dan 36\sqrt{3} ni ayirish.

$$x=\frac{-36\sqrt{3}-6}{18}$$

-6-36\sqrt{3} ni 18 ga bo'lish.

$$x=-2\sqrt{3}-\frac{1}{3}$$

Tenglama yechildi.

$$x=2\sqrt{3}-\frac{1}{3} x=-2\sqrt{3}-\frac{1}{3}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -\frac{1}{3} teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 3\left(3x+1\right)^{2} ga, \left(1+3x\right)^{2},3 ning eng kichik karralisiga ko‘paytiring.

$$-3\left(-36\right)=\left(3x+1\right)^{2}$$

108 hosil qilish uchun -3 va -36 ni ko'paytirish.

$$108=\left(3x+1\right)^{2}$$

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

$$108=9x^{2}+6x+1$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$9x^{2}+6x+1=108$$

Ikkala tarafdan 1 ni ayirish.

$$9x^{2}+6x=108-1$$

107 olish uchun 108 dan 1 ni ayirish.

$$9x^{2}+6x=107$$

Ikki tarafini 9 ga bo‘ling.

$$\frac{9x^{2}+6x}{9}=\frac{107}{9}$$

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

$$x^{2}+\frac{6}{9}x=\frac{107}{9}$$

\frac{6}{9} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{2}{3}x=\frac{107}{9}$$

\frac{2}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{3} olish uchun. Keyin, \frac{1}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{2}{3}x+\left(\frac{1}{3}\right)^{2}=\frac{107}{9}+\left(\frac{1}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{3} kvadratini chiqarish.

$$x^{2}+\frac{2}{3}x+\frac{1}{9}=\frac{107+1}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{107}{9} ni \frac{1}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{2}{3}x+\frac{1}{9}=12$$

x^{2}+\frac{2}{3}x+\frac{1}{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.

$$\left(x+\frac{1}{3}\right)^{2}=12$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{3}\right)^{2}}=\sqrt{12}$$

Qisqartirish.

$$x+\frac{1}{3}=2\sqrt{3} x+\frac{1}{3}=-2\sqrt{3}$$

Tenglamaning ikkala tarafidan \frac{1}{3} ni ayirish.

$$x=2\sqrt{3}-\frac{1}{3} x=-2\sqrt{3}-\frac{1}{3}$$