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

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

x uchun yechish (complex solution)

$$x=\frac{7+\sqrt{11}i}{10}\approx 0,7+0,331662479i$$

$$x=\frac{-\sqrt{11}i+7}{10}\approx 0,7-0,331662479i$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

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

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

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

-7 kvadratini chiqarish.

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

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-20\times 3}}{2\times 5}$$

-20 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-60}}{2\times 5}$$

49 ni -60 ga qo'shish.

$$x=\frac{-\left(-7\right)±\sqrt{-11}}{2\times 5}$$

-11 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-7\right)±\sqrt{11}i}{2\times 5}$$

-7 ning teskarisi 7 ga teng.

$$x=\frac{7±\sqrt{11}i}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{7±\sqrt{11}i}{10}$$

x=\frac{7±\sqrt{11}i}{10} tenglamasini yeching, bunda ± musbat. 7 ni i\sqrt{11} ga qo'shish.

$$x=\frac{7+\sqrt{11}i}{10}$$

x=\frac{7±\sqrt{11}i}{10} tenglamasini yeching, bunda ± manfiy. 7 dan i\sqrt{11} ni ayirish.

$$x=\frac{-\sqrt{11}i+7}{10}$$

Tenglama yechildi.

$$x=\frac{7+\sqrt{11}i}{10} x=\frac{-\sqrt{11}i+7}{10}$$

Kvadratni yakunlash bo'yicha qadamlar

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

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

Ikkala tarafdan 3 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

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

Ikki tarafini 5 ga bo‘ling.

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

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

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

-\frac{7}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{10} olish uchun. Keyin, -\frac{7}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{7}{5}x+\left(-\frac{7}{10}\right)^{2}=-\frac{3}{5}+\left(-\frac{7}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{10} kvadratini chiqarish.

$$x^{2}-\frac{7}{5}x+\frac{49}{100}=-\frac{3}{5}+\frac{49}{100}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{3}{5} ni \frac{49}{100} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{7}{5}x+\frac{49}{100}=-\frac{11}{100}$$

x^{2}-\frac{7}{5}x+\frac{49}{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.

$$\left(x-\frac{7}{10}\right)^{2}=-\frac{11}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{10}\right)^{2}}=\sqrt{-\frac{11}{100}}$$

Qisqartirish.

$$x-\frac{7}{10}=\frac{\sqrt{11}i}{10} x-\frac{7}{10}=-\frac{\sqrt{11}i}{10}$$

\frac{7}{10} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{7+\sqrt{11}i}{10} x=\frac{-\sqrt{11}i+7}{10}$$