5x^2-3x+1=0 eching

$$5 { x }^{ 2 } -3x+1=0$$

x uchun yechish (complex solution)

$$x=\frac{3+\sqrt{11}i}{10}\approx 0,3+0,331662479i$$

$$x=\frac{-\sqrt{11}i+3}{10}\approx 0,3-0,331662479i$$

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.

$$5x^{2}-3x+1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, -3 ni b va 1 ni c bilan almashtiring.

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

-3 kvadratini chiqarish.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 5}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9-20}}{2\times 5}$$

9 ni -20 ga qo'shish.

$$x=\frac{-\left(-3\right)±\sqrt{-11}}{2\times 5}$$

-11 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-3\right)±\sqrt{11}i}{2\times 5}$$

-3 ning teskarisi 3 ga teng.

$$x=\frac{3±\sqrt{11}i}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{3±\sqrt{11}i}{10}$$

x=\frac{3±\sqrt{11}i}{10} tenglamasini yeching, bunda ± musbat. 3 ni i\sqrt{11} ga qo'shish.

$$x=\frac{3+\sqrt{11}i}{10}$$

x=\frac{3±\sqrt{11}i}{10} tenglamasini yeching, bunda ± manfiy. 3 dan i\sqrt{11} ni ayirish.

$$x=\frac{-\sqrt{11}i+3}{10}$$

Tenglama yechildi.

$$x=\frac{3+\sqrt{11}i}{10} x=\frac{-\sqrt{11}i+3}{10}$$

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.

$$5x^{2}-3x+1=0$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$5x^{2}-3x+1-1=-1$$

O‘zidan 1 ayirilsa 0 qoladi.

$$5x^{2}-3x=-1$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5x^{2}-3x}{5}=-\frac{1}{5}$$

5 ga bo'lish 5 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{3}{5}x=-\frac{1}{5}$$

-\frac{3}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{10} olish uchun. Keyin, -\frac{3}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{3}{5}x+\left(-\frac{3}{10}\right)^{2}=-\frac{1}{5}+\left(-\frac{3}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{10} kvadratini chiqarish.

$$x^{2}-\frac{3}{5}x+\frac{9}{100}=-\frac{1}{5}+\frac{9}{100}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{5} ni \frac{9}{100} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{3}{5}x+\frac{9}{100}=-\frac{11}{100}$$

x^{2}-\frac{3}{5}x+\frac{9}{100} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{3}{10}\right)^{2}=-\frac{11}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{10}\right)^{2}}=\sqrt{-\frac{11}{100}}$$

Qisqartirish.

$$x-\frac{3}{10}=\frac{\sqrt{11}i}{10} x-\frac{3}{10}=-\frac{\sqrt{11}i}{10}$$

\frac{3}{10} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{3+\sqrt{11}i}{10} x=\frac{-\sqrt{11}i+3}{10}$$