-5x-2=3x^2-4x eching

$$-5x-2=3 { x }^{ 2 } -4x$$

x uchun yechish (complex solution)

$$x=\frac{-\sqrt{23}i-1}{6}\approx -0,166666667-0,799305254i$$

$$x=\frac{-1+\sqrt{23}i}{6}\approx -0,166666667+0,799305254i$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 3x^{2} ni ayirish.

$$-5x-2-3x^{2}=-4x$$

4x ni ikki tarafga qo’shing.

$$-5x-2-3x^{2}+4x=0$$

-x ni olish uchun -5x va 4x ni birlashtirish.

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

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.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, -1 ni b va -2 ni c bilan almashtiring.

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

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1+12\left(-2\right)}}{2\left(-3\right)}$$

12 ni -2 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1-24}}{2\left(-3\right)}$$

1 ni -24 ga qo'shish.

$$x=\frac{-\left(-1\right)±\sqrt{-23}}{2\left(-3\right)}$$

-23 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-1\right)±\sqrt{23}i}{2\left(-3\right)}$$

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±\sqrt{23}i}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{1±\sqrt{23}i}{-6}$$

x=\frac{1±\sqrt{23}i}{-6} tenglamasini yeching, bunda ± musbat. 1 ni i\sqrt{23} ga qo'shish.

$$x=\frac{1+\sqrt{23}i}{-6}$$

1+i\sqrt{23} ni -6 ga bo'lish.

$$x=\frac{-\sqrt{23}i-1}{6}$$

x=\frac{1±\sqrt{23}i}{-6} tenglamasini yeching, bunda ± manfiy. 1 dan i\sqrt{23} ni ayirish.

$$x=\frac{-\sqrt{23}i+1}{-6}$$

1-i\sqrt{23} ni -6 ga bo'lish.

$$x=\frac{-1+\sqrt{23}i}{6}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{23}i-1}{6} x=\frac{-1+\sqrt{23}i}{6}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 3x^{2} ni ayirish.

$$-5x-2-3x^{2}=-4x$$

4x ni ikki tarafga qo’shing.

$$-5x-2-3x^{2}+4x=0$$

-x ni olish uchun -5x va 4x ni birlashtirish.

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

2 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

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

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

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

Ikki tarafini -3 ga bo‘ling.

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

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

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

-1 ni -3 ga bo'lish.

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

2 ni -3 ga bo'lish.

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

\frac{1}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{6} olish uchun. Keyin, \frac{1}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{1}{3}x+\left(\frac{1}{6}\right)^{2}=-\frac{2}{3}+\left(\frac{1}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{6} kvadratini chiqarish.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=-\frac{2}{3}+\frac{1}{36}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{2}{3} ni \frac{1}{36} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=-\frac{23}{36}$$

x^{2}+\frac{1}{3}x+\frac{1}{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.

$$\left(x+\frac{1}{6}\right)^{2}=-\frac{23}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{6}\right)^{2}}=\sqrt{-\frac{23}{36}}$$

Qisqartirish.

$$x+\frac{1}{6}=\frac{\sqrt{23}i}{6} x+\frac{1}{6}=-\frac{\sqrt{23}i}{6}$$

Tenglamaning ikkala tarafidan \frac{1}{6} ni ayirish.

$$x=\frac{-1+\sqrt{23}i}{6} x=\frac{-\sqrt{23}i-1}{6}$$