quad9a^2-6a-1=0 eching

$$\quad 9 a ^ { 2 } - 6 a - 1 = 0$$

a uchun yechish

$$a=\frac{\sqrt{2}+1}{3}\approx 0,804737854$$

$$a=\frac{1-\sqrt{2}}{3}\approx -0,138071187$$

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.

$$9a^{2}-6a-1=0$$

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 -1 ni c bilan almashtiring.

$$a=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 9\left(-1\right)}}{2\times 9}$$

-6 kvadratini chiqarish.

$$a=\frac{-\left(-6\right)±\sqrt{36-4\times 9\left(-1\right)}}{2\times 9}$$

-4 ni 9 marotabaga ko'paytirish.

$$a=\frac{-\left(-6\right)±\sqrt{36-36\left(-1\right)}}{2\times 9}$$

-36 ni -1 marotabaga ko'paytirish.

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

36 ni 36 ga qo'shish.

$$a=\frac{-\left(-6\right)±\sqrt{72}}{2\times 9}$$

72 ning kvadrat ildizini chiqarish.

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

-6 ning teskarisi 6 ga teng.

$$a=\frac{6±6\sqrt{2}}{2\times 9}$$

2 ni 9 marotabaga ko'paytirish.

$$a=\frac{6±6\sqrt{2}}{18}$$

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

$$a=\frac{6\sqrt{2}+6}{18}$$

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

$$a=\frac{\sqrt{2}+1}{3}$$

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

$$a=\frac{6-6\sqrt{2}}{18}$$

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

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

Tenglama yechildi.

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

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.

$$9a^{2}-6a-1=0$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$9a^{2}-6a-1-\left(-1\right)=-\left(-1\right)$$

O‘zidan -1 ayirilsa 0 qoladi.

$$9a^{2}-6a=-\left(-1\right)$$

0 dan -1 ni ayirish.

$$9a^{2}-6a=1$$

Ikki tarafini 9 ga bo‘ling.

$$\frac{9a^{2}-6a}{9}=\frac{1}{9}$$

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

$$a^{2}+\left(-\frac{6}{9}\right)a=\frac{1}{9}$$

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

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

-\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.

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

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

$$a^{2}-\frac{2}{3}a+\frac{1}{9}=\frac{1+1}{9}$$

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

$$a^{2}-\frac{2}{3}a+\frac{1}{9}=\frac{2}{9}$$

a^{2}-\frac{2}{3}a+\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.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

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

\frac{1}{3} ni tenglamaning ikkala tarafiga qo'shish.

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