x^2/9-4/3x=-2 eching

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

x uchun yechish

$$x=3\sqrt{2}+6\approx 10,242640687$$

$$x=6-3\sqrt{2}\approx 1,757359313$$

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.

$$\frac{1}{9}x^{2}-\frac{4}{3}x=-2$$

2 ni tenglamaning ikkala tarafiga qo'shish.

$$\frac{1}{9}x^{2}-\frac{4}{3}x-\left(-2\right)=-2-\left(-2\right)$$

O‘zidan -2 ayirilsa 0 qoladi.

$$\frac{1}{9}x^{2}-\frac{4}{3}x-\left(-2\right)=0$$

0 dan -2 ni ayirish.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} \frac{1}{9} ni a, -\frac{4}{3} ni b va 2 ni c bilan almashtiring.

$$x=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\left(-\frac{4}{3}\right)^{2}-4\times \frac{1}{9}\times 2}}{2\times \frac{1}{9}}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{4}{3} kvadratini chiqarish.

$$x=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\frac{16}{9}-4\times \frac{1}{9}\times 2}}{2\times \frac{1}{9}}$$

-4 ni \frac{1}{9} marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\frac{16}{9}-\frac{4}{9}\times 2}}{2\times \frac{1}{9}}$$

-\frac{4}{9} ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\frac{16-8}{9}}}{2\times \frac{1}{9}}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{16}{9} ni -\frac{8}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\frac{8}{9}}}{2\times \frac{1}{9}}$$

\frac{8}{9} ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-\frac{4}{3}\right)±\frac{2\sqrt{2}}{3}}{2\times \frac{1}{9}}$$

-\frac{4}{3} ning teskarisi \frac{4}{3} ga teng.

$$x=\frac{\frac{4}{3}±\frac{2\sqrt{2}}{3}}{2\times \frac{1}{9}}$$

2 ni \frac{1}{9} marotabaga ko'paytirish.

$$x=\frac{\frac{4}{3}±\frac{2\sqrt{2}}{3}}{\frac{2}{9}}$$

x=\frac{\frac{4}{3}±\frac{2\sqrt{2}}{3}}{\frac{2}{9}} tenglamasini yeching, bunda ± musbat. \frac{4}{3} ni \frac{2\sqrt{2}}{3} ga qo'shish.

$$x=\frac{2\sqrt{2}+4}{\frac{2}{9}\times 3}$$

\frac{4+2\sqrt{2}}{3} ni \frac{2}{9} ga bo'lish \frac{4+2\sqrt{2}}{3} ga k'paytirish \frac{2}{9} ga qaytarish.

$$x=3\sqrt{2}+6$$

x=\frac{\frac{4}{3}±\frac{2\sqrt{2}}{3}}{\frac{2}{9}} tenglamasini yeching, bunda ± manfiy. \frac{4}{3} dan \frac{2\sqrt{2}}{3} ni ayirish.

$$x=\frac{4-2\sqrt{2}}{\frac{2}{9}\times 3}$$

\frac{4-2\sqrt{2}}{3} ni \frac{2}{9} ga bo'lish \frac{4-2\sqrt{2}}{3} ga k'paytirish \frac{2}{9} ga qaytarish.

$$x=6-3\sqrt{2}$$

Tenglama yechildi.

$$x=3\sqrt{2}+6 x=6-3\sqrt{2}$$

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.

$$\frac{1}{9}x^{2}-\frac{4}{3}x=-2$$

Ikkala tarafini 9 ga ko‘paytiring.

$$\frac{\frac{1}{9}x^{2}-\frac{4}{3}x}{\frac{1}{9}}=-\frac{2}{\frac{1}{9}}$$

\frac{1}{9} ga bo'lish \frac{1}{9} ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{\frac{4}{3}}{\frac{1}{9}}\right)x=-\frac{2}{\frac{1}{9}}$$

-\frac{4}{3} ni \frac{1}{9} ga bo'lish -\frac{4}{3} ga k'paytirish \frac{1}{9} ga qaytarish.

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

-2 ni \frac{1}{9} ga bo'lish -2 ga k'paytirish \frac{1}{9} ga qaytarish.

$$x^{2}-12x=-18$$

-12 ni bo‘lish, x shartining koeffitsienti, 2 ga -6 olish uchun. Keyin, -6 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-12x+\left(-6\right)^{2}=-18+\left(-6\right)^{2}$$

-6 kvadratini chiqarish.

$$x^{2}-12x+36=-18+36$$

-18 ni 36 ga qo'shish.

$$x^{2}-12x+36=18$$

x^{2}-12x+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.

$$\left(x-6\right)^{2}=18$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{18}$$

Qisqartirish.

$$x-6=3\sqrt{2} x-6=-3\sqrt{2}$$

6 ni tenglamaning ikkala tarafiga qo'shish.

$$x=3\sqrt{2}+6 x=6-3\sqrt{2}$$