31x^2-3x+1=0 eching

$$31 x ^ { 2 } - 3 x + 1 = 0$$

x uchun yechish (complex solution)

$$x=\frac{3+\sqrt{115}i}{62}\approx 0,048387097+0,172964602i$$

$$x=\frac{-\sqrt{115}i+3}{62}\approx 0,048387097-0,172964602i$$

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.

$$31x^{2}-3x+1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 31 ni a, -3 ni b va 1 ni c bilan almashtiring.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 31}}{2\times 31}$$

-3 kvadratini chiqarish.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 31}}{2\times 31}$$

-4 ni 31 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9-124}}{2\times 31}$$

9 ni -124 ga qo'shish.

$$x=\frac{-\left(-3\right)±\sqrt{-115}}{2\times 31}$$

-115 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-3\right)±\sqrt{115}i}{2\times 31}$$

-3 ning teskarisi 3 ga teng.

$$x=\frac{3±\sqrt{115}i}{2\times 31}$$

2 ni 31 marotabaga ko'paytirish.

$$x=\frac{3±\sqrt{115}i}{62}$$

x=\frac{3±\sqrt{115}i}{62} tenglamasini yeching, bunda ± musbat. 3 ni i\sqrt{115} ga qo'shish.

$$x=\frac{3+\sqrt{115}i}{62}$$

x=\frac{3±\sqrt{115}i}{62} tenglamasini yeching, bunda ± manfiy. 3 dan i\sqrt{115} ni ayirish.

$$x=\frac{-\sqrt{115}i+3}{62}$$

Tenglama yechildi.

$$x=\frac{3+\sqrt{115}i}{62} x=\frac{-\sqrt{115}i+3}{62}$$

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.

$$31x^{2}-3x+1=0$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$31x^{2}-3x+1-1=-1$$

O‘zidan 1 ayirilsa 0 qoladi.

$$31x^{2}-3x=-1$$

Ikki tarafini 31 ga bo‘ling.

$$\frac{31x^{2}-3x}{31}=-\frac{1}{31}$$

31 ga bo'lish 31 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{3}{31}x=-\frac{1}{31}$$

-\frac{3}{31} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{62} olish uchun. Keyin, -\frac{3}{62} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{3}{31}x+\left(-\frac{3}{62}\right)^{2}=-\frac{1}{31}+\left(-\frac{3}{62}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{62} kvadratini chiqarish.

$$x^{2}-\frac{3}{31}x+\frac{9}{3844}=-\frac{1}{31}+\frac{9}{3844}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{31} ni \frac{9}{3844} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{3}{31}x+\frac{9}{3844}=-\frac{115}{3844}$$

x^{2}-\frac{3}{31}x+\frac{9}{3844} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{3}{62}\right)^{2}=-\frac{115}{3844}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{62}\right)^{2}}=\sqrt{-\frac{115}{3844}}$$

Qisqartirish.

$$x-\frac{3}{62}=\frac{\sqrt{115}i}{62} x-\frac{3}{62}=-\frac{\sqrt{115}i}{62}$$

\frac{3}{62} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{3+\sqrt{115}i}{62} x=\frac{-\sqrt{115}i+3}{62}$$