7x^2-12x+8=0 eching

$$7 x ^ { 2 } - 12 x + 8 = 0$$

x uchun yechish (complex solution)

$$x=\frac{6+2\sqrt{5}i}{7}\approx 0,857142857+0,638876565i$$

$$x=\frac{-2\sqrt{5}i+6}{7}\approx 0,857142857-0,638876565i$$

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.

$$7x^{2}-12x+8=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 7 ni a, -12 ni b va 8 ni c bilan almashtiring.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 7\times 8}}{2\times 7}$$

-12 kvadratini chiqarish.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 7\times 8}}{2\times 7}$$

-4 ni 7 marotabaga ko'paytirish.

$$x=\frac{-\left(-12\right)±\sqrt{144-28\times 8}}{2\times 7}$$

-28 ni 8 marotabaga ko'paytirish.

$$x=\frac{-\left(-12\right)±\sqrt{144-224}}{2\times 7}$$

144 ni -224 ga qo'shish.

$$x=\frac{-\left(-12\right)±\sqrt{-80}}{2\times 7}$$

-80 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-12\right)±4\sqrt{5}i}{2\times 7}$$

-12 ning teskarisi 12 ga teng.

$$x=\frac{12±4\sqrt{5}i}{2\times 7}$$

2 ni 7 marotabaga ko'paytirish.

$$x=\frac{12±4\sqrt{5}i}{14}$$

x=\frac{12±4\sqrt{5}i}{14} tenglamasini yeching, bunda ± musbat. 12 ni 4i\sqrt{5} ga qo'shish.

$$x=\frac{12+4\sqrt{5}i}{14}$$

12+4i\sqrt{5} ni 14 ga bo'lish.

$$x=\frac{6+2\sqrt{5}i}{7}$$

x=\frac{12±4\sqrt{5}i}{14} tenglamasini yeching, bunda ± manfiy. 12 dan 4i\sqrt{5} ni ayirish.

$$x=\frac{-4\sqrt{5}i+12}{14}$$

12-4i\sqrt{5} ni 14 ga bo'lish.

$$x=\frac{-2\sqrt{5}i+6}{7}$$

Tenglama yechildi.

$$x=\frac{6+2\sqrt{5}i}{7} x=\frac{-2\sqrt{5}i+6}{7}$$

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.

$$7x^{2}-12x+8=0$$

Tenglamaning ikkala tarafidan 8 ni ayirish.

$$7x^{2}-12x+8-8=-8$$

O‘zidan 8 ayirilsa 0 qoladi.

$$7x^{2}-12x=-8$$

Ikki tarafini 7 ga bo‘ling.

$$\frac{7x^{2}-12x}{7}=-\frac{8}{7}$$

7 ga bo'lish 7 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{12}{7}x=-\frac{8}{7}$$

-\frac{12}{7} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{6}{7} olish uchun. Keyin, -\frac{6}{7} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{12}{7}x+\left(-\frac{6}{7}\right)^{2}=-\frac{8}{7}+\left(-\frac{6}{7}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{6}{7} kvadratini chiqarish.

$$x^{2}-\frac{12}{7}x+\frac{36}{49}=-\frac{8}{7}+\frac{36}{49}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{8}{7} ni \frac{36}{49} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{12}{7}x+\frac{36}{49}=-\frac{20}{49}$$

x^{2}-\frac{12}{7}x+\frac{36}{49} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{6}{7}\right)^{2}=-\frac{20}{49}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{6}{7}\right)^{2}}=\sqrt{-\frac{20}{49}}$$

Qisqartirish.

$$x-\frac{6}{7}=\frac{2\sqrt{5}i}{7} x-\frac{6}{7}=-\frac{2\sqrt{5}i}{7}$$

\frac{6}{7} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{6+2\sqrt{5}i}{7} x=\frac{-2\sqrt{5}i+6}{7}$$