b+1=-1/2(b+1)^2 eching

$$b + 1 = - \frac { 1 } { 2 } ( b + 1 ) ^ { 2 }$$

b uchun yechish

$$b=-3$$

$$b=-1$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(b+1\right)^{2} kengaytirilishi uchun ishlating.

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

-\frac{1}{2} ga b^{2}+2b+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

\frac{1}{2}b^{2} ni ikki tarafga qo’shing.

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

b ni ikki tarafga qo’shing.

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

2b ni olish uchun b va b ni birlashtirish.

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

\frac{1}{2} ni ikki tarafga qo’shing.

$$2b+1+\frac{1}{2}b^{2}+\frac{1}{2}=0$$

\frac{3}{2} olish uchun 1 va \frac{1}{2}'ni qo'shing.

$$2b+\frac{3}{2}+\frac{1}{2}b^{2}=0$$

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.

$$\frac{1}{2}b^{2}+2b+\frac{3}{2}=0$$

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

$$b=\frac{-2±\sqrt{2^{2}-4\times \frac{1}{2}\times \frac{3}{2}}}{2\times \frac{1}{2}}$$

2 kvadratini chiqarish.

$$b=\frac{-2±\sqrt{4-4\times \frac{1}{2}\times \frac{3}{2}}}{2\times \frac{1}{2}}$$

-4 ni \frac{1}{2} marotabaga ko'paytirish.

$$b=\frac{-2±\sqrt{4-2\times \frac{3}{2}}}{2\times \frac{1}{2}}$$

-2 ni \frac{3}{2} marotabaga ko'paytirish.

$$b=\frac{-2±\sqrt{4-3}}{2\times \frac{1}{2}}$$

4 ni -3 ga qo'shish.

$$b=\frac{-2±\sqrt{1}}{2\times \frac{1}{2}}$$

1 ning kvadrat ildizini chiqarish.

$$b=\frac{-2±1}{2\times \frac{1}{2}}$$

2 ni \frac{1}{2} marotabaga ko'paytirish.

$$b=\frac{-2±1}{1}$$

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

$$b=-\frac{1}{1}$$

-1 ni 1 ga bo'lish.

$$b=-1$$

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

$$b=-\frac{3}{1}$$

-3 ni 1 ga bo'lish.

$$b=-3$$

Tenglama yechildi.

$$b=-1 b=-3$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(b+1\right)^{2} kengaytirilishi uchun ishlating.

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

-\frac{1}{2} ga b^{2}+2b+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

\frac{1}{2}b^{2} ni ikki tarafga qo’shing.

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

b ni ikki tarafga qo’shing.

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

2b ni olish uchun b va b ni birlashtirish.

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

Ikkala tarafdan 1 ni ayirish.

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

-\frac{3}{2} olish uchun -\frac{1}{2} dan 1 ni ayirish.

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

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

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

Ikkala tarafini 2 ga ko‘paytiring.

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

\frac{1}{2} ga bo'lish \frac{1}{2} ga ko'paytirishni bekor qiladi.

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

2 ni \frac{1}{2} ga bo'lish 2 ga k'paytirish \frac{1}{2} ga qaytarish.

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

-\frac{3}{2} ni \frac{1}{2} ga bo'lish -\frac{3}{2} ga k'paytirish \frac{1}{2} ga qaytarish.

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

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

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

2 kvadratini chiqarish.

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

-3 ni 4 ga qo'shish.

$$b^{2}+4b+4=1$$

b^{2}+4b+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.

$$\left(b+2\right)^{2}=1$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(b+2\right)^{2}}=\sqrt{1}$$

Qisqartirish.

$$b+2=1 b+2=-1$$

Tenglamaning ikkala tarafidan 2 ni ayirish.

$$b=-1 b=-3$$