9y^2-12y+2=0 eching

$$9 y ^ { 2 } - 12 y + 2 = 0$$

y uchun yechish

$$y = \frac{\sqrt{2} + 2}{3} \approx 1,138071187$$

$$y=\frac{2-\sqrt{2}}{3}\approx 0,195262146$$

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.

$$9y^{2}-12y+2=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 9 ni a, -12 ni b va 2 ni c bilan almashtiring.

$$y=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 9\times 2}}{2\times 9}$$

-12 kvadratini chiqarish.

$$y=\frac{-\left(-12\right)±\sqrt{144-4\times 9\times 2}}{2\times 9}$$

-4 ni 9 marotabaga ko'paytirish.

$$y=\frac{-\left(-12\right)±\sqrt{144-36\times 2}}{2\times 9}$$

-36 ni 2 marotabaga ko'paytirish.

$$y=\frac{-\left(-12\right)±\sqrt{144-72}}{2\times 9}$$

144 ni -72 ga qo'shish.

$$y=\frac{-\left(-12\right)±\sqrt{72}}{2\times 9}$$

72 ning kvadrat ildizini chiqarish.

$$y=\frac{-\left(-12\right)±6\sqrt{2}}{2\times 9}$$

-12 ning teskarisi 12 ga teng.

$$y=\frac{12±6\sqrt{2}}{2\times 9}$$

2 ni 9 marotabaga ko'paytirish.

$$y=\frac{12±6\sqrt{2}}{18}$$

y=\frac{12±6\sqrt{2}}{18} tenglamasini yeching, bunda ± musbat. 12 ni 6\sqrt{2} ga qo'shish.

$$y=\frac{6\sqrt{2}+12}{18}$$

12+6\sqrt{2} ni 18 ga bo'lish.

$$y=\frac{\sqrt{2}+2}{3}$$

y=\frac{12±6\sqrt{2}}{18} tenglamasini yeching, bunda ± manfiy. 12 dan 6\sqrt{2} ni ayirish.

$$y=\frac{12-6\sqrt{2}}{18}$$

12-6\sqrt{2} ni 18 ga bo'lish.

$$y=\frac{2-\sqrt{2}}{3}$$

Tenglama yechildi.

$$y=\frac{\sqrt{2}+2}{3} y=\frac{2-\sqrt{2}}{3}$$

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.

$$9y^{2}-12y+2=0$$

Tenglamaning ikkala tarafidan 2 ni ayirish.

$$9y^{2}-12y+2-2=-2$$

O‘zidan 2 ayirilsa 0 qoladi.

$$9y^{2}-12y=-2$$

Ikki tarafini 9 ga bo‘ling.

$$\frac{9y^{2}-12y}{9}=-\frac{2}{9}$$

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

$$y^{2}+\left(-\frac{12}{9}\right)y=-\frac{2}{9}$$

\frac{-12}{9} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

-\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.

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

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

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

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{2}{9} ni \frac{4}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

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

y^{2}-\frac{4}{3}y+\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.

$$\left(y-\frac{2}{3}\right)^{2}=\frac{2}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(y-\frac{2}{3}\right)^{2}}=\sqrt{\frac{2}{9}}$$

Qisqartirish.

$$y-\frac{2}{3}=\frac{\sqrt{2}}{3} y-\frac{2}{3}=-\frac{\sqrt{2}}{3}$$

\frac{2}{3} ni tenglamaning ikkala tarafiga qo'shish.

$$y=\frac{\sqrt{2}+2}{3} y=\frac{2-\sqrt{2}}{3}$$