x=x^2+1 eching

$$x = x ^ { 2 } + 1$$

x uchun yechish (complex solution)

$$x=\frac{-\sqrt{3}i+1}{2}\approx 0,5-0,866025404i$$

$$x=\frac{1+\sqrt{3}i}{2}\approx 0,5+0,866025404i$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan x^{2} ni ayirish.

$$x-x^{2}=1$$

Ikkala tarafdan 1 ni ayirish.

$$x-x^{2}-1=0$$

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.

$$-x^{2}+x-1=0$$

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

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

1 kvadratini chiqarish.

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

-4 ni -1 marotabaga ko'paytirish.

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

4 ni -1 marotabaga ko'paytirish.

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

1 ni -4 ga qo'shish.

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

-3 ning kvadrat ildizini chiqarish.

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

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{3}i}{-2}$$

x=\frac{-1±\sqrt{3}i}{-2} tenglamasini yeching, bunda ± musbat. -1 ni i\sqrt{3} ga qo'shish.

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

-1+i\sqrt{3} ni -2 ga bo'lish.

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

x=\frac{-1±\sqrt{3}i}{-2} tenglamasini yeching, bunda ± manfiy. -1 dan i\sqrt{3} ni ayirish.

$$x=\frac{-\sqrt{3}i-1}{-2}$$

-1-i\sqrt{3} ni -2 ga bo'lish.

$$x=\frac{1+\sqrt{3}i}{2}$$

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan x^{2} ni ayirish.

$$x-x^{2}=1$$

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

$$-x^{2}+x=1$$

Ikki tarafini -1 ga bo‘ling.

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

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

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

1 ni -1 ga bo'lish.

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

1 ni -1 ga bo'lish.

$$x^{2}-x=-1$$

-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.

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

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

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

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

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

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.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x-\frac{1}{2}=\frac{\sqrt{3}i}{2} x-\frac{1}{2}=-\frac{\sqrt{3}i}{2}$$

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

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