10x^2-x+3=0 eching

$$10 x ^ { 2 } - x + 3 = 0$$

x uchun yechish (complex solution)

$$x=\frac{1+\sqrt{119}i}{20}\approx 0,05+0,545435606i$$

$$x=\frac{-\sqrt{119}i+1}{20}\approx 0,05-0,545435606i$$

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.

$$10x^{2}-x+3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 10 ni a, -1 ni b va 3 ni c bilan almashtiring.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 10\times 3}}{2\times 10}$$

-4 ni 10 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1-40\times 3}}{2\times 10}$$

-40 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1-120}}{2\times 10}$$

1 ni -120 ga qo'shish.

$$x=\frac{-\left(-1\right)±\sqrt{-119}}{2\times 10}$$

-119 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-1\right)±\sqrt{119}i}{2\times 10}$$

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±\sqrt{119}i}{2\times 10}$$

2 ni 10 marotabaga ko'paytirish.

$$x=\frac{1±\sqrt{119}i}{20}$$

x=\frac{1±\sqrt{119}i}{20} tenglamasini yeching, bunda ± musbat. 1 ni i\sqrt{119} ga qo'shish.

$$x=\frac{1+\sqrt{119}i}{20}$$

x=\frac{1±\sqrt{119}i}{20} tenglamasini yeching, bunda ± manfiy. 1 dan i\sqrt{119} ni ayirish.

$$x=\frac{-\sqrt{119}i+1}{20}$$

Tenglama yechildi.

$$x=\frac{1+\sqrt{119}i}{20} x=\frac{-\sqrt{119}i+1}{20}$$

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.

$$10x^{2}-x+3=0$$

Tenglamaning ikkala tarafidan 3 ni ayirish.

$$10x^{2}-x+3-3=-3$$

O‘zidan 3 ayirilsa 0 qoladi.

$$10x^{2}-x=-3$$

Ikki tarafini 10 ga bo‘ling.

$$\frac{10x^{2}-x}{10}=-\frac{3}{10}$$

10 ga bo'lish 10 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{1}{10}x=-\frac{3}{10}$$

-\frac{1}{10} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{20} olish uchun. Keyin, -\frac{1}{20} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1}{10}x+\left(-\frac{1}{20}\right)^{2}=-\frac{3}{10}+\left(-\frac{1}{20}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{20} kvadratini chiqarish.

$$x^{2}-\frac{1}{10}x+\frac{1}{400}=-\frac{3}{10}+\frac{1}{400}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{3}{10} ni \frac{1}{400} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{10}x+\frac{1}{400}=-\frac{119}{400}$$

x^{2}-\frac{1}{10}x+\frac{1}{400} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{20}\right)^{2}=-\frac{119}{400}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{20}\right)^{2}}=\sqrt{-\frac{119}{400}}$$

Qisqartirish.

$$x-\frac{1}{20}=\frac{\sqrt{119}i}{20} x-\frac{1}{20}=-\frac{\sqrt{119}i}{20}$$

\frac{1}{20} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{1+\sqrt{119}i}{20} x=\frac{-\sqrt{119}i+1}{20}$$