3x^2-7x+5=0 eching

$$3 x ^ { 2 } - 7 x + 5 = 0$$

x uchun yechish (complex solution)

$$x=\frac{7+\sqrt{11}i}{6}\approx 1,166666667+0,552770798i$$

$$x=\frac{-\sqrt{11}i+7}{6}\approx 1,166666667-0,552770798i$$

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.

$$3x^{2}-7x+5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -7 ni b va 5 ni c bilan almashtiring.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 3\times 5}}{2\times 3}$$

-7 kvadratini chiqarish.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 3\times 5}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-12\times 5}}{2\times 3}$$

-12 ni 5 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-60}}{2\times 3}$$

49 ni -60 ga qo'shish.

$$x=\frac{-\left(-7\right)±\sqrt{-11}}{2\times 3}$$

-11 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-7\right)±\sqrt{11}i}{2\times 3}$$

-7 ning teskarisi 7 ga teng.

$$x=\frac{7±\sqrt{11}i}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{7±\sqrt{11}i}{6}$$

x=\frac{7±\sqrt{11}i}{6} tenglamasini yeching, bunda ± musbat. 7 ni i\sqrt{11} ga qo'shish.

$$x=\frac{7+\sqrt{11}i}{6}$$

x=\frac{7±\sqrt{11}i}{6} tenglamasini yeching, bunda ± manfiy. 7 dan i\sqrt{11} ni ayirish.

$$x=\frac{-\sqrt{11}i+7}{6}$$

Tenglama yechildi.

$$x=\frac{7+\sqrt{11}i}{6} x=\frac{-\sqrt{11}i+7}{6}$$

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.

$$3x^{2}-7x+5=0$$

Tenglamaning ikkala tarafidan 5 ni ayirish.

$$3x^{2}-7x+5-5=-5$$

O‘zidan 5 ayirilsa 0 qoladi.

$$3x^{2}-7x=-5$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}-7x}{3}=-\frac{5}{3}$$

3 ga bo'lish 3 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{7}{3}x=-\frac{5}{3}$$

-\frac{7}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{6} olish uchun. Keyin, -\frac{7}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{7}{3}x+\left(-\frac{7}{6}\right)^{2}=-\frac{5}{3}+\left(-\frac{7}{6}\right)^{2}$$

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

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=-\frac{5}{3}+\frac{49}{36}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{5}{3} ni \frac{49}{36} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=-\frac{11}{36}$$

x^{2}-\frac{7}{3}x+\frac{49}{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.

$$\left(x-\frac{7}{6}\right)^{2}=-\frac{11}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{6}\right)^{2}}=\sqrt{-\frac{11}{36}}$$

Qisqartirish.

$$x-\frac{7}{6}=\frac{\sqrt{11}i}{6} x-\frac{7}{6}=-\frac{\sqrt{11}i}{6}$$

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

$$x=\frac{7+\sqrt{11}i}{6} x=\frac{-\sqrt{11}i+7}{6}$$