13a^2-12a-9=0 eching

$$13 a ^ { 2 } - 12 a - 9 = 0$$

a uchun yechish

$$a = \frac{3 \sqrt{17} + 6}{13} \approx 1,413024375$$

$$a=\frac{6-3\sqrt{17}}{13}\approx -0,489947452$$

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.

$$13a^{2}-12a-9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 13 ni a, -12 ni b va -9 ni c bilan almashtiring.

$$a=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 13\left(-9\right)}}{2\times 13}$$

-12 kvadratini chiqarish.

$$a=\frac{-\left(-12\right)±\sqrt{144-4\times 13\left(-9\right)}}{2\times 13}$$

-4 ni 13 marotabaga ko'paytirish.

$$a=\frac{-\left(-12\right)±\sqrt{144-52\left(-9\right)}}{2\times 13}$$

-52 ni -9 marotabaga ko'paytirish.

$$a=\frac{-\left(-12\right)±\sqrt{144+468}}{2\times 13}$$

144 ni 468 ga qo'shish.

$$a=\frac{-\left(-12\right)±\sqrt{612}}{2\times 13}$$

612 ning kvadrat ildizini chiqarish.

$$a=\frac{-\left(-12\right)±6\sqrt{17}}{2\times 13}$$

-12 ning teskarisi 12 ga teng.

$$a=\frac{12±6\sqrt{17}}{2\times 13}$$

2 ni 13 marotabaga ko'paytirish.

$$a=\frac{12±6\sqrt{17}}{26}$$

a=\frac{12±6\sqrt{17}}{26} tenglamasini yeching, bunda ± musbat. 12 ni 6\sqrt{17} ga qo'shish.

$$a=\frac{6\sqrt{17}+12}{26}$$

12+6\sqrt{17} ni 26 ga bo'lish.

$$a=\frac{3\sqrt{17}+6}{13}$$

a=\frac{12±6\sqrt{17}}{26} tenglamasini yeching, bunda ± manfiy. 12 dan 6\sqrt{17} ni ayirish.

$$a=\frac{12-6\sqrt{17}}{26}$$

12-6\sqrt{17} ni 26 ga bo'lish.

$$a=\frac{6-3\sqrt{17}}{13}$$

Tenglama yechildi.

$$a=\frac{3\sqrt{17}+6}{13} a=\frac{6-3\sqrt{17}}{13}$$

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.

$$13a^{2}-12a-9=0$$

9 ni tenglamaning ikkala tarafiga qo'shish.

$$13a^{2}-12a-9-\left(-9\right)=-\left(-9\right)$$

O‘zidan -9 ayirilsa 0 qoladi.

$$13a^{2}-12a=-\left(-9\right)$$

0 dan -9 ni ayirish.

$$13a^{2}-12a=9$$

Ikki tarafini 13 ga bo‘ling.

$$\frac{13a^{2}-12a}{13}=\frac{9}{13}$$

13 ga bo'lish 13 ga ko'paytirishni bekor qiladi.

$$a^{2}-\frac{12}{13}a=\frac{9}{13}$$

-\frac{12}{13} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{6}{13} olish uchun. Keyin, -\frac{6}{13} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$a^{2}-\frac{12}{13}a+\left(-\frac{6}{13}\right)^{2}=\frac{9}{13}+\left(-\frac{6}{13}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{6}{13} kvadratini chiqarish.

$$a^{2}-\frac{12}{13}a+\frac{36}{169}=\frac{9}{13}+\frac{36}{169}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{9}{13} ni \frac{36}{169} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$a^{2}-\frac{12}{13}a+\frac{36}{169}=\frac{153}{169}$$

a^{2}-\frac{12}{13}a+\frac{36}{169} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(a-\frac{6}{13}\right)^{2}=\frac{153}{169}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(a-\frac{6}{13}\right)^{2}}=\sqrt{\frac{153}{169}}$$

Qisqartirish.

$$a-\frac{6}{13}=\frac{3\sqrt{17}}{13} a-\frac{6}{13}=-\frac{3\sqrt{17}}{13}$$

\frac{6}{13} ni tenglamaning ikkala tarafiga qo'shish.

$$a=\frac{3\sqrt{17}+6}{13} a=\frac{6-3\sqrt{17}}{13}$$