3x^2=7x-10 eching

$$3 x ^ { 2 } = 7 x - 10$$

x uchun yechish (complex solution)

$$x=\frac{7+\sqrt{71}i}{6}\approx 1,166666667+1,404358296i$$

$$x=\frac{-\sqrt{71}i+7}{6}\approx 1,166666667-1,404358296i$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 7x ni ayirish.

$$3x^{2}-7x=-10$$

10 ni ikki tarafga qo’shing.

$$3x^{2}-7x+10=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -7 ni b va 10 ni c bilan almashtiring.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 3\times 10}}{2\times 3}$$

-7 kvadratini chiqarish.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 3\times 10}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-12\times 10}}{2\times 3}$$

-12 ni 10 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-120}}{2\times 3}$$

49 ni -120 ga qo'shish.

$$x=\frac{-\left(-7\right)±\sqrt{-71}}{2\times 3}$$

-71 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-7\right)±\sqrt{71}i}{2\times 3}$$

-7 ning teskarisi 7 ga teng.

$$x=\frac{7±\sqrt{71}i}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{7±\sqrt{71}i}{6}$$

x=\frac{7±\sqrt{71}i}{6} tenglamasini yeching, bunda ± musbat. 7 ni i\sqrt{71} ga qo'shish.

$$x=\frac{7+\sqrt{71}i}{6}$$

x=\frac{7±\sqrt{71}i}{6} tenglamasini yeching, bunda ± manfiy. 7 dan i\sqrt{71} ni ayirish.

$$x=\frac{-\sqrt{71}i+7}{6}$$

Tenglama yechildi.

$$x=\frac{7+\sqrt{71}i}{6} x=\frac{-\sqrt{71}i+7}{6}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 7x ni ayirish.

$$3x^{2}-7x=-10$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}-7x}{3}=-\frac{10}{3}$$

3 ga bo'lish 3 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{7}{3}x=-\frac{10}{3}$$

-\frac{7}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{6} olish uchun. Keyin, -\frac{7}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{7}{3}x+\left(-\frac{7}{6}\right)^{2}=-\frac{10}{3}+\left(-\frac{7}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{6} kvadratini chiqarish.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=-\frac{10}{3}+\frac{49}{36}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{10}{3} ni \frac{49}{36} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=-\frac{71}{36}$$

x^{2}-\frac{7}{3}x+\frac{49}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{7}{6}\right)^{2}=-\frac{71}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{6}\right)^{2}}=\sqrt{-\frac{71}{36}}$$

Qisqartirish.

$$x-\frac{7}{6}=\frac{\sqrt{71}i}{6} x-\frac{7}{6}=-\frac{\sqrt{71}i}{6}$$

\frac{7}{6} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{7+\sqrt{71}i}{6} x=\frac{-\sqrt{71}i+7}{6}$$