2x-3x^2-4=0 eching

$$2 x - 3 x ^ { 2 } - 4 = 0$$

x uchun yechish (complex solution)

$$x=\frac{-\sqrt{11}i+1}{3}\approx 0,333333333-1,105541597i$$

$$x=\frac{1+\sqrt{11}i}{3}\approx 0,333333333+1,105541597i$$

Kvadrat tenglamalaridan foydalanish qoidalari

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$-3x^{2}+2x-4=0$$

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

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

2 kvadratini chiqarish.

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

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+12\left(-4\right)}}{2\left(-3\right)}$$

12 ni -4 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4-48}}{2\left(-3\right)}$$

4 ni -48 ga qo'shish.

$$x=\frac{-2±\sqrt{-44}}{2\left(-3\right)}$$

-44 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±2\sqrt{11}i}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{-2±2\sqrt{11}i}{-6}$$

x=\frac{-2±2\sqrt{11}i}{-6} tenglamasini yeching, bunda ± musbat. -2 ni 2i\sqrt{11} ga qo'shish.

$$x=\frac{-2+2\sqrt{11}i}{-6}$$

-2+2i\sqrt{11} ni -6 ga bo'lish.

$$x=\frac{-\sqrt{11}i+1}{3}$$

x=\frac{-2±2\sqrt{11}i}{-6} tenglamasini yeching, bunda ± manfiy. -2 dan 2i\sqrt{11} ni ayirish.

$$x=\frac{-2\sqrt{11}i-2}{-6}$$

-2-2i\sqrt{11} ni -6 ga bo'lish.

$$x=\frac{1+\sqrt{11}i}{3}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{11}i+1}{3} x=\frac{1+\sqrt{11}i}{3}$$

Kvadratni yakunlash bo'yicha qadamlar

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-3x^{2}+2x-4=0$$

4 ni tenglamaning ikkala tarafiga qo'shish.

$$-3x^{2}+2x-4-\left(-4\right)=-\left(-4\right)$$

O‘zidan -4 ayirilsa 0 qoladi.

$$-3x^{2}+2x=-\left(-4\right)$$

0 dan -4 ni ayirish.

$$-3x^{2}+2x=4$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3x^{2}+2x}{-3}=\frac{4}{-3}$$

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

$$x^{2}+\frac{2}{-3}x=\frac{4}{-3}$$

2 ni -3 ga bo'lish.

$$x^{2}-\frac{2}{3}x=\frac{4}{-3}$$

4 ni -3 ga bo'lish.

$$x^{2}-\frac{2}{3}x=-\frac{4}{3}$$

-\frac{2}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{3} olish uchun. Keyin, -\frac{1}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{2}{3}x+\left(-\frac{1}{3}\right)^{2}=-\frac{4}{3}+\left(-\frac{1}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{3} kvadratini chiqarish.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=-\frac{4}{3}+\frac{1}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{4}{3} ni \frac{1}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=-\frac{11}{9}$$

x^{2}-\frac{2}{3}x+\frac{1}{9} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{3}\right)^{2}=-\frac{11}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{3}\right)^{2}}=\sqrt{-\frac{11}{9}}$$

Qisqartirish.

$$x-\frac{1}{3}=\frac{\sqrt{11}i}{3} x-\frac{1}{3}=-\frac{\sqrt{11}i}{3}$$

\frac{1}{3} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{1+\sqrt{11}i}{3} x=\frac{-\sqrt{11}i+1}{3}$$