5x^2=3x-7 eching

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

x uchun yechish (complex solution)

$$x=\frac{3+\sqrt{131}i}{10}\approx 0,3+1,144552314i$$

$$x=\frac{-\sqrt{131}i+3}{10}\approx 0,3-1,144552314i$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 3x ni ayirish.

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

7 ni ikki tarafga qo’shing.

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

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 7 ni c bilan almashtiring.

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

-3 kvadratini chiqarish.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 5\times 7}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9-20\times 7}}{2\times 5}$$

-20 ni 7 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9-140}}{2\times 5}$$

9 ni -140 ga qo'shish.

$$x=\frac{-\left(-3\right)±\sqrt{-131}}{2\times 5}$$

-131 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-3\right)±\sqrt{131}i}{2\times 5}$$

-3 ning teskarisi 3 ga teng.

$$x=\frac{3±\sqrt{131}i}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{3±\sqrt{131}i}{10}$$

x=\frac{3±\sqrt{131}i}{10} tenglamasini yeching, bunda ± musbat. 3 ni i\sqrt{131} ga qo'shish.

$$x=\frac{3+\sqrt{131}i}{10}$$

x=\frac{3±\sqrt{131}i}{10} tenglamasini yeching, bunda ± manfiy. 3 dan i\sqrt{131} ni ayirish.

$$x=\frac{-\sqrt{131}i+3}{10}$$

Tenglama yechildi.

$$x=\frac{3+\sqrt{131}i}{10} x=\frac{-\sqrt{131}i+3}{10}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 3x ni ayirish.

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

Ikki tarafini 5 ga bo‘ling.

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

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

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

-\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.

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

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

$$x^{2}-\frac{3}{5}x+\frac{9}{100}=-\frac{7}{5}+\frac{9}{100}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{7}{5} ni \frac{9}{100} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{3}{5}x+\frac{9}{100}=-\frac{131}{100}$$

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.

$$\left(x-\frac{3}{10}\right)^{2}=-\frac{131}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{10}\right)^{2}}=\sqrt{-\frac{131}{100}}$$

Qisqartirish.

$$x-\frac{3}{10}=\frac{\sqrt{131}i}{10} x-\frac{3}{10}=-\frac{\sqrt{131}i}{10}$$

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

$$x=\frac{3+\sqrt{131}i}{10} x=\frac{-\sqrt{131}i+3}{10}$$