1/2left(1-k/2right)left(2-kright)=left(k+2right)left(1-k/2right) eching

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

k uchun yechish

$$k=2$$

$$k=-\frac{2}{3}\approx -0,666666667$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 2 ga ko'paytirish.

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

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

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

1-\frac{k}{2} ifodaning har bir elementini 2-k ifodaning har bir elementiga ko‘paytirish orqali taqsimot qonuni xususiyatlarini qo‘llash mumkin.

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

2\left(-\frac{k}{2}\right) ni yagona kasrga aylantiring.

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

2 va 2 ni qisqartiring.

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

-2k ni olish uchun -k va -k ni birlashtirish.

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

1 hosil qilish uchun -1 va -1 ni ko'paytirish.

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

\frac{k}{2}k ni yagona kasrga aylantiring.

$$2-2k+\frac{kk}{2}=2\left(k+2\right)\left(1-\frac{k}{2}\right)$$

k^{2} hosil qilish uchun k va k ni ko'paytirish.

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

2 ga k+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2-2k+\frac{k^{2}}{2}=\left(2k+4\right)\left(1-\frac{k}{2}\right)$$

2k+4 ifodaning har bir elementini 1-\frac{k}{2} ifodaning har bir elementiga ko‘paytirish orqali taqsimot qonuni xususiyatlarini qo‘llash mumkin.

$$2-2k+\frac{k^{2}}{2}=2k+2k\left(-\frac{k}{2}\right)+4+4\left(-\frac{k}{2}\right)$$

2\left(-\frac{k}{2}\right) ni yagona kasrga aylantiring.

$$2-2k+\frac{k^{2}}{2}=2k+\frac{-2k}{2}k+4+4\left(-\frac{k}{2}\right)$$

2 va 2 ni qisqartiring.

$$2-2k+\frac{k^{2}}{2}=2k-kk+4+4\left(-\frac{k}{2}\right)$$

4 va 2 ichida eng katta umumiy 2 faktorini bekor qiling.

$$2-2k+\frac{k^{2}}{2}=2k-kk+4-2k$$

0 ni olish uchun 2k va -2k ni birlashtirish.

$$2-2k+\frac{k^{2}}{2}=-kk+4$$

k^{2} hosil qilish uchun k va k ni ko'paytirish.

$$2-2k+\frac{k^{2}}{2}=-k^{2}+4$$

k^{2} ni ikki tarafga qo’shing.

$$2-2k+\frac{k^{2}}{2}+k^{2}=4$$

\frac{3}{2}k^{2} ni olish uchun \frac{k^{2}}{2} va k^{2} ni birlashtirish.

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

Ikkala tarafdan 4 ni ayirish.

$$2-2k+\frac{3}{2}k^{2}-4=0$$

-2 olish uchun 2 dan 4 ni ayirish.

$$-2-2k+\frac{3}{2}k^{2}=0$$

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.

$$\frac{3}{2}k^{2}-2k-2=0$$

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

$$k=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times \frac{3}{2}\left(-2\right)}}{2\times \frac{3}{2}}$$

-2 kvadratini chiqarish.

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

-4 ni \frac{3}{2} marotabaga ko'paytirish.

$$k=\frac{-\left(-2\right)±\sqrt{4-6\left(-2\right)}}{2\times \frac{3}{2}}$$

-6 ni -2 marotabaga ko'paytirish.

$$k=\frac{-\left(-2\right)±\sqrt{4+12}}{2\times \frac{3}{2}}$$

4 ni 12 ga qo'shish.

$$k=\frac{-\left(-2\right)±\sqrt{16}}{2\times \frac{3}{2}}$$

16 ning kvadrat ildizini chiqarish.

$$k=\frac{-\left(-2\right)±4}{2\times \frac{3}{2}}$$

-2 ning teskarisi 2 ga teng.

$$k=\frac{2±4}{2\times \frac{3}{2}}$$

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

$$k=\frac{2±4}{3}$$

k=\frac{2±4}{3} tenglamasini yeching, bunda ± musbat. 2 ni 4 ga qo'shish.

$$k=\frac{6}{3}$$

6 ni 3 ga bo'lish.

$$k=2$$

k=\frac{2±4}{3} tenglamasini yeching, bunda ± manfiy. 2 dan 4 ni ayirish.

$$k=-\frac{2}{3}$$

Tenglama yechildi.

$$k=2 k=-\frac{2}{3}$$

Kvadratni yakunlash bo'yicha qadamlar

Tenglamaning ikkala tarafini 2 ga ko'paytirish.

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

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

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

1-\frac{k}{2} ifodaning har bir elementini 2-k ifodaning har bir elementiga ko‘paytirish orqali taqsimot qonuni xususiyatlarini qo‘llash mumkin.

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

2\left(-\frac{k}{2}\right) ni yagona kasrga aylantiring.

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

2 va 2 ni qisqartiring.

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

-2k ni olish uchun -k va -k ni birlashtirish.

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

1 hosil qilish uchun -1 va -1 ni ko'paytirish.

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

\frac{k}{2}k ni yagona kasrga aylantiring.

$$2-2k+\frac{kk}{2}=2\left(k+2\right)\left(1-\frac{k}{2}\right)$$

k^{2} hosil qilish uchun k va k ni ko'paytirish.

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

2 ga k+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2-2k+\frac{k^{2}}{2}=\left(2k+4\right)\left(1-\frac{k}{2}\right)$$

2k+4 ifodaning har bir elementini 1-\frac{k}{2} ifodaning har bir elementiga ko‘paytirish orqali taqsimot qonuni xususiyatlarini qo‘llash mumkin.

$$2-2k+\frac{k^{2}}{2}=2k+2k\left(-\frac{k}{2}\right)+4+4\left(-\frac{k}{2}\right)$$

2\left(-\frac{k}{2}\right) ni yagona kasrga aylantiring.

$$2-2k+\frac{k^{2}}{2}=2k+\frac{-2k}{2}k+4+4\left(-\frac{k}{2}\right)$$

2 va 2 ni qisqartiring.

$$2-2k+\frac{k^{2}}{2}=2k-kk+4+4\left(-\frac{k}{2}\right)$$

4 va 2 ichida eng katta umumiy 2 faktorini bekor qiling.

$$2-2k+\frac{k^{2}}{2}=2k-kk+4-2k$$

0 ni olish uchun 2k va -2k ni birlashtirish.

$$2-2k+\frac{k^{2}}{2}=-kk+4$$

k^{2} hosil qilish uchun k va k ni ko'paytirish.

$$2-2k+\frac{k^{2}}{2}=-k^{2}+4$$

k^{2} ni ikki tarafga qo’shing.

$$2-2k+\frac{k^{2}}{2}+k^{2}=4$$

\frac{3}{2}k^{2} ni olish uchun \frac{k^{2}}{2} va k^{2} ni birlashtirish.

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

Ikkala tarafdan 2 ni ayirish.

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

2 olish uchun 4 dan 2 ni ayirish.

$$-2k+\frac{3}{2}k^{2}=2$$

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

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

Tenglamaning ikki tarafini \frac{3}{2} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

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

\frac{3}{2} ga bo'lish \frac{3}{2} ga ko'paytirishni bekor qiladi.

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

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

$$k^{2}-\frac{4}{3}k=\frac{2}{\frac{3}{2}}$$

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

$$k^{2}-\frac{4}{3}k=\frac{4}{3}$$

-\frac{4}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{2}{3} olish uchun. Keyin, -\frac{2}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$k^{2}-\frac{4}{3}k+\left(-\frac{2}{3}\right)^{2}=\frac{4}{3}+\left(-\frac{2}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{2}{3} kvadratini chiqarish.

$$k^{2}-\frac{4}{3}k+\frac{4}{9}=\frac{4}{3}+\frac{4}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{4}{3} ni \frac{4}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$k^{2}-\frac{4}{3}k+\frac{4}{9}=\frac{16}{9}$$

k^{2}-\frac{4}{3}k+\frac{4}{9} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(k-\frac{2}{3}\right)^{2}=\frac{16}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(k-\frac{2}{3}\right)^{2}}=\sqrt{\frac{16}{9}}$$

Qisqartirish.

$$k-\frac{2}{3}=\frac{4}{3} k-\frac{2}{3}=-\frac{4}{3}$$

\frac{2}{3} ni tenglamaning ikkala tarafiga qo'shish.

$$k=2 k=-\frac{2}{3}$$