2x-2x^2=12 eching

$$2x-2 { x }^{ 2 } = 12$$

x uchun yechish (complex solution)

$$x=\frac{1+\sqrt{23}i}{2}\approx 0,5+2,397915762i$$

$$x=\frac{-\sqrt{23}i+1}{2}\approx 0,5-2,397915762i$$

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.

$$-2x^{2}+2x=12$$

Tenglamaning ikkala tarafidan 12 ni ayirish.

$$-2x^{2}+2x-12=12-12$$

O‘zidan 12 ayirilsa 0 qoladi.

$$-2x^{2}+2x-12=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -2 ni a, 2 ni b va -12 ni c bilan almashtiring.

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

2 kvadratini chiqarish.

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

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+8\left(-12\right)}}{2\left(-2\right)}$$

8 ni -12 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4-96}}{2\left(-2\right)}$$

4 ni -96 ga qo'shish.

$$x=\frac{-2±\sqrt{-92}}{2\left(-2\right)}$$

-92 ning kvadrat ildizini chiqarish.

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

2 ni -2 marotabaga ko'paytirish.

$$x=\frac{-2±2\sqrt{23}i}{-4}$$

x=\frac{-2±2\sqrt{23}i}{-4} tenglamasini yeching, bunda ± musbat. -2 ni 2i\sqrt{23} ga qo'shish.

$$x=\frac{-2+2\sqrt{23}i}{-4}$$

-2+2i\sqrt{23} ni -4 ga bo'lish.

$$x=\frac{-\sqrt{23}i+1}{2}$$

x=\frac{-2±2\sqrt{23}i}{-4} tenglamasini yeching, bunda ± manfiy. -2 dan 2i\sqrt{23} ni ayirish.

$$x=\frac{-2\sqrt{23}i-2}{-4}$$

-2-2i\sqrt{23} ni -4 ga bo'lish.

$$x=\frac{1+\sqrt{23}i}{2}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{23}i+1}{2} x=\frac{1+\sqrt{23}i}{2}$$

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.

$$-2x^{2}+2x=12$$

Ikki tarafini -2 ga bo‘ling.

$$\frac{-2x^{2}+2x}{-2}=\frac{12}{-2}$$

-2 ga bo'lish -2 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{2}{-2}x=\frac{12}{-2}$$

2 ni -2 ga bo'lish.

$$x^{2}-x=\frac{12}{-2}$$

12 ni -2 ga bo'lish.

$$x^{2}-x=-6$$

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{2} kvadratini chiqarish.

$$x^{2}-x+\frac{1}{4}=-6+\frac{1}{4}$$

-6 ni \frac{1}{4} ga qo'shish.

$$x^{2}-x+\frac{1}{4}=-\frac{23}{4}$$

x^{2}-x+\frac{1}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{2}\right)^{2}=-\frac{23}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{23}{4}}$$

Qisqartirish.

$$x-\frac{1}{2}=\frac{\sqrt{23}i}{2} x-\frac{1}{2}=-\frac{\sqrt{23}i}{2}$$

\frac{1}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{1+\sqrt{23}i}{2} x=\frac{-\sqrt{23}i+1}{2}$$