f(x)=3x^2-5x-2quad0quadg(x)=2x-7 eching

$$f ( x ) = 3 x ^ { 2 } - 5 x - 2 \quad 0 \quad g ( x ) = 2 x - 7$$

x uchun yechish (complex solution)

$$x=\frac{7+\sqrt{35}i}{6}$$

$$x=\frac{-\sqrt{35}i+7}{6}$$

Kvadrat tenglamalaridan foydalanish qoidalari

0 hosil qilish uchun 2 va 0 ni ko'paytirish.

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

Har qanday sonni nolga ko‘paytirsangiz, nol chiqadi.

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

Ikkala tarafdan 2x ni ayirish.

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

7 ni ikki tarafga qo’shing.

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

Shartlarni qayta saralash.

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

-7x ni olish uchun -5x va -2x ni birlashtirish.

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

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 7 ni c bilan almashtiring.

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

-7 kvadratini chiqarish.

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

-4 ni 3 marotabaga ko'paytirish.

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

-12 ni 7 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-84}}{2\times 3}$$

49 ni -84 ga qo'shish.

$$x=\frac{-\left(-7\right)±\sqrt{-35}}{2\times 3}$$

-35 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-7\right)±\sqrt{35}i}{2\times 3}$$

-7 ning teskarisi 7 ga teng.

$$x=\frac{7±\sqrt{35}i}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{7±\sqrt{35}i}{6}$$

x=\frac{7±\sqrt{35}i}{6} tenglamasini yeching, bunda ± musbat. 7 ni i\sqrt{35} ga qo'shish.

$$x=\frac{7+\sqrt{35}i}{6}$$

x=\frac{7±\sqrt{35}i}{6} tenglamasini yeching, bunda ± manfiy. 7 dan i\sqrt{35} ni ayirish.

$$x=\frac{-\sqrt{35}i+7}{6}$$

Tenglama yechildi.

$$x=\frac{7+\sqrt{35}i}{6} x=\frac{-\sqrt{35}i+7}{6}$$

Kvadratni yakunlash bo'yicha qadamlar

0 hosil qilish uchun 2 va 0 ni ko'paytirish.

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

Har qanday sonni nolga ko‘paytirsangiz, nol chiqadi.

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

Ikkala tarafdan 2x ni ayirish.

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

Shartlarni qayta saralash.

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

-7x ni olish uchun -5x va -2x ni birlashtirish.

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

Ikki tarafini 3 ga bo‘ling.

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

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

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

-\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.

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

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

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

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{7}{3} ni \frac{49}{36} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=-\frac{35}{36}$$

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.

$$\left(x-\frac{7}{6}\right)^{2}=-\frac{35}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{6}\right)^{2}}=\sqrt{-\frac{35}{36}}$$

Qisqartirish.

$$x-\frac{7}{6}=\frac{\sqrt{35}i}{6} x-\frac{7}{6}=-\frac{\sqrt{35}i}{6}$$

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

$$x=\frac{7+\sqrt{35}i}{6} x=\frac{-\sqrt{35}i+7}{6}$$

g uchun yechish (complex solution)

$$g\in \mathrm{C}$$

$$x=\frac{7+\sqrt{35}i}{6}\text{ or }x=\frac{-\sqrt{35}i+7}{6}$$