5x^2-x+7=0 eching

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

x uchun yechish (complex solution)

$$x=\frac{1+\sqrt{139}i}{10}\approx 0,1+1,178982612i$$

$$x=\frac{-\sqrt{139}i+1}{10}\approx 0,1-1,178982612i$$

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.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, -1 ni b va 7 ni c bilan almashtiring.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 5\times 7}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1-20\times 7}}{2\times 5}$$

-20 ni 7 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1-140}}{2\times 5}$$

1 ni -140 ga qo'shish.

$$x=\frac{-\left(-1\right)±\sqrt{-139}}{2\times 5}$$

-139 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-1\right)±\sqrt{139}i}{2\times 5}$$

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±\sqrt{139}i}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{1±\sqrt{139}i}{10}$$

x=\frac{1±\sqrt{139}i}{10} tenglamasini yeching, bunda ± musbat. 1 ni i\sqrt{139} ga qo'shish.

$$x=\frac{1+\sqrt{139}i}{10}$$

x=\frac{1±\sqrt{139}i}{10} tenglamasini yeching, bunda ± manfiy. 1 dan i\sqrt{139} ni ayirish.

$$x=\frac{-\sqrt{139}i+1}{10}$$

Tenglama yechildi.

$$x=\frac{1+\sqrt{139}i}{10} x=\frac{-\sqrt{139}i+1}{10}$$

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.

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

Tenglamaning ikkala tarafidan 7 ni ayirish.

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

O‘zidan 7 ayirilsa 0 qoladi.

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

Ikki tarafini 5 ga bo‘ling.

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

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

$$x^{2}-\frac{1}{5}x=-\frac{7}{5}$$

-\frac{1}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{10} olish uchun. Keyin, -\frac{1}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1}{5}x+\left(-\frac{1}{10}\right)^{2}=-\frac{7}{5}+\left(-\frac{1}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{10} kvadratini chiqarish.

$$x^{2}-\frac{1}{5}x+\frac{1}{100}=-\frac{7}{5}+\frac{1}{100}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{7}{5} ni \frac{1}{100} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{5}x+\frac{1}{100}=-\frac{139}{100}$$

x^{2}-\frac{1}{5}x+\frac{1}{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.

$$\left(x-\frac{1}{10}\right)^{2}=-\frac{139}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{10}\right)^{2}}=\sqrt{-\frac{139}{100}}$$

Qisqartirish.

$$x-\frac{1}{10}=\frac{\sqrt{139}i}{10} x-\frac{1}{10}=-\frac{\sqrt{139}i}{10}$$

\frac{1}{10} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{1+\sqrt{139}i}{10} x=\frac{-\sqrt{139}i+1}{10}$$