n+1=n^2-1 eching

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

n uchun yechish

$$n=-1$$

$$n=2$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan n^{2} ni ayirish.

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

1 ni ikki tarafga qo’shing.

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

2 olish uchun 1 va 1'ni qo'shing.

$$n+2-n^{2}=0$$

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$-n^{2}+n+2=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -n^{2}+an+bn+2 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=1 ab=-2=-2$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. Faqat bundan juftlik tizim yechimidir.

$$a=2 b=-1$$

-n^{2}+n+2 ni \left(-n^{2}+2n\right)+\left(-n+2\right) sifatida qaytadan yozish.

$$\left(-n^{2}+2n\right)+\left(-n+2\right)$$

Birinchi guruhda -n ni va ikkinchi guruhda -1 ni faktordan chiqaring.

$$-n\left(n-2\right)-\left(n-2\right)$$

Distributiv funktsiyasidan foydalangan holda n-2 umumiy terminini chiqaring.

$$\left(n-2\right)\left(-n-1\right)$$

Tenglamani yechish uchun n-2=0 va -n-1=0 ni yeching.

$$n=2 n=-1$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan n^{2} ni ayirish.

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

1 ni ikki tarafga qo’shing.

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

2 olish uchun 1 va 1'ni qo'shing.

$$n+2-n^{2}=0$$

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.

$$-n^{2}+n+2=0$$

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 2 ni c bilan almashtiring.

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

1 kvadratini chiqarish.

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

-4 ni -1 marotabaga ko'paytirish.

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

4 ni 2 marotabaga ko'paytirish.

$$n=\frac{-1±\sqrt{1+8}}{2\left(-1\right)}$$

1 ni 8 ga qo'shish.

$$n=\frac{-1±\sqrt{9}}{2\left(-1\right)}$$

9 ning kvadrat ildizini chiqarish.

$$n=\frac{-1±3}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$n=\frac{-1±3}{-2}$$

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

$$n=\frac{2}{-2}$$

2 ni -2 ga bo'lish.

$$n=-1$$

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

$$n=-\frac{4}{-2}$$

-4 ni -2 ga bo'lish.

$$n=2$$

Tenglama yechildi.

$$n=-1 n=2$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan n^{2} ni ayirish.

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

Ikkala tarafdan 1 ni ayirish.

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

-2 olish uchun -1 dan 1 ni ayirish.

$$n-n^{2}=-2$$

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

$$-n^{2}+n=-2$$

Ikki tarafini -1 ga bo‘ling.

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

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

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

1 ni -1 ga bo'lish.

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

-2 ni -1 ga bo'lish.

$$n^{2}-n=2$$

-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.

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

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

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

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

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

n^{2}-n+\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.

$$\left(n-\frac{1}{2}\right)^{2}=\frac{9}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(n-\frac{1}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Qisqartirish.

$$n-\frac{1}{2}=\frac{3}{2} n-\frac{1}{2}=-\frac{3}{2}$$

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

$$n=2 n=-1$$