-3x^2-3x+11=2x eching

$$- 3 x ^ { 2 } - 3 x + 11 = 2 x$$

x uchun yechish

$$x = \frac{\sqrt{157} - 5}{6} \approx 1,254994014$$

$$x=\frac{-\sqrt{157}-5}{6}\approx -2,921660681$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 2x ni ayirish.

$$-3x^{2}-3x+11-2x=0$$

-5x ni olish uchun -3x va -2x ni birlashtirish.

$$-3x^{2}-5x+11=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, -5 ni b va 11 ni c bilan almashtiring.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-3\right)\times 11}}{2\left(-3\right)}$$

-5 kvadratini chiqarish.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-3\right)\times 11}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-\left(-5\right)±\sqrt{25+12\times 11}}{2\left(-3\right)}$$

12 ni 11 marotabaga ko'paytirish.

$$x=\frac{-\left(-5\right)±\sqrt{25+132}}{2\left(-3\right)}$$

25 ni 132 ga qo'shish.

$$x=\frac{-\left(-5\right)±\sqrt{157}}{2\left(-3\right)}$$

-5 ning teskarisi 5 ga teng.

$$x=\frac{5±\sqrt{157}}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{5±\sqrt{157}}{-6}$$

x=\frac{5±\sqrt{157}}{-6} tenglamasini yeching, bunda ± musbat. 5 ni \sqrt{157} ga qo'shish.

$$x=\frac{\sqrt{157}+5}{-6}$$

5+\sqrt{157} ni -6 ga bo'lish.

$$x=\frac{-\sqrt{157}-5}{6}$$

x=\frac{5±\sqrt{157}}{-6} tenglamasini yeching, bunda ± manfiy. 5 dan \sqrt{157} ni ayirish.

$$x=\frac{5-\sqrt{157}}{-6}$$

5-\sqrt{157} ni -6 ga bo'lish.

$$x=\frac{\sqrt{157}-5}{6}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{157}-5}{6} x=\frac{\sqrt{157}-5}{6}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 2x ni ayirish.

$$-3x^{2}-3x+11-2x=0$$

-5x ni olish uchun -3x va -2x ni birlashtirish.

$$-3x^{2}-5x+11=0$$

Ikkala tarafdan 11 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$-3x^{2}-5x=-11$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3x^{2}-5x}{-3}=-\frac{11}{-3}$$

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

$$x^{2}+\left(-\frac{5}{-3}\right)x=-\frac{11}{-3}$$

-5 ni -3 ga bo'lish.

$$x^{2}+\frac{5}{3}x=-\frac{11}{-3}$$

-11 ni -3 ga bo'lish.

$$x^{2}+\frac{5}{3}x=\frac{11}{3}$$

\frac{5}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{5}{6} olish uchun. Keyin, \frac{5}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{5}{3}x+\left(\frac{5}{6}\right)^{2}=\frac{11}{3}+\left(\frac{5}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{5}{6} kvadratini chiqarish.

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=\frac{11}{3}+\frac{25}{36}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{11}{3} ni \frac{25}{36} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=\frac{157}{36}$$

x^{2}+\frac{5}{3}x+\frac{25}{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.

$$\left(x+\frac{5}{6}\right)^{2}=\frac{157}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{5}{6}\right)^{2}}=\sqrt{\frac{157}{36}}$$

Qisqartirish.

$$x+\frac{5}{6}=\frac{\sqrt{157}}{6} x+\frac{5}{6}=-\frac{\sqrt{157}}{6}$$

Tenglamaning ikkala tarafidan \frac{5}{6} ni ayirish.

$$x=\frac{\sqrt{157}-5}{6} x=\frac{-\sqrt{157}-5}{6}$$