2v^2-10v+44=(v-7)^2 eching

$$2 v ^ { 2 } - 10 v + 44 = ( v - 7 ) ^ { 2 }$$

v uchun yechish

$$v=-5$$

$$v=1$$

Faktorlash qadamlari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(v-7\right)^{2} kengaytirilishi uchun ishlating.

$$2v^{2}-10v+44=v^{2}-14v+49$$

Ikkala tarafdan v^{2} ni ayirish.

$$2v^{2}-10v+44-v^{2}=-14v+49$$

v^{2} ni olish uchun 2v^{2} va -v^{2} ni birlashtirish.

$$v^{2}-10v+44=-14v+49$$

14v ni ikki tarafga qo’shing.

$$v^{2}-10v+44+14v=49$$

4v ni olish uchun -10v va 14v ni birlashtirish.

$$v^{2}+4v+44=49$$

Ikkala tarafdan 49 ni ayirish.

$$v^{2}+4v+44-49=0$$

-5 olish uchun 44 dan 49 ni ayirish.

$$v^{2}+4v-5=0$$

Bu tenglamani yechish uchun v^{2}+\left(a+b\right)v+ab=\left(v+a\right)\left(v+b\right) formulasi yordamida v^{2}+4v-5 ni faktorlang. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=4 ab=-5$$

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.

$$a=-1 b=5$$

Faktorlangan \left(v+a\right)\left(v+b\right) ifodani olingan qiymatlar bilan qaytadan yozing.

$$\left(v-1\right)\left(v+5\right)$$

Tenglamani yechish uchun v-1=0 va v+5=0 ni yeching.

$$v=1 v=-5$$

Guruhlash orqali faktorlash qadamlari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(v-7\right)^{2} kengaytirilishi uchun ishlating.

$$2v^{2}-10v+44=v^{2}-14v+49$$

Ikkala tarafdan v^{2} ni ayirish.

$$2v^{2}-10v+44-v^{2}=-14v+49$$

v^{2} ni olish uchun 2v^{2} va -v^{2} ni birlashtirish.

$$v^{2}-10v+44=-14v+49$$

14v ni ikki tarafga qo’shing.

$$v^{2}-10v+44+14v=49$$

4v ni olish uchun -10v va 14v ni birlashtirish.

$$v^{2}+4v+44=49$$

Ikkala tarafdan 49 ni ayirish.

$$v^{2}+4v+44-49=0$$

-5 olish uchun 44 dan 49 ni ayirish.

$$v^{2}+4v-5=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon v^{2}+av+bv-5 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=4 ab=1\left(-5\right)=-5$$

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.

$$a=-1 b=5$$

v^{2}+4v-5 ni \left(v^{2}-v\right)+\left(5v-5\right) sifatida qaytadan yozish.

$$\left(v^{2}-v\right)+\left(5v-5\right)$$

Birinchi guruhda v ni va ikkinchi guruhda 5 ni faktordan chiqaring.

$$v\left(v-1\right)+5\left(v-1\right)$$

Distributiv funktsiyasidan foydalangan holda v-1 umumiy terminini chiqaring.

$$\left(v-1\right)\left(v+5\right)$$

Tenglamani yechish uchun v-1=0 va v+5=0 ni yeching.

$$v=1 v=-5$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(v-7\right)^{2} kengaytirilishi uchun ishlating.

$$2v^{2}-10v+44=v^{2}-14v+49$$

Ikkala tarafdan v^{2} ni ayirish.

$$2v^{2}-10v+44-v^{2}=-14v+49$$

v^{2} ni olish uchun 2v^{2} va -v^{2} ni birlashtirish.

$$v^{2}-10v+44=-14v+49$$

14v ni ikki tarafga qo’shing.

$$v^{2}-10v+44+14v=49$$

4v ni olish uchun -10v va 14v ni birlashtirish.

$$v^{2}+4v+44=49$$

Ikkala tarafdan 49 ni ayirish.

$$v^{2}+4v+44-49=0$$

-5 olish uchun 44 dan 49 ni ayirish.

$$v^{2}+4v-5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 4 ni b va -5 ni c bilan almashtiring.

$$v=\frac{-4±\sqrt{4^{2}-4\left(-5\right)}}{2}$$

4 kvadratini chiqarish.

$$v=\frac{-4±\sqrt{16-4\left(-5\right)}}{2}$$

-4 ni -5 marotabaga ko'paytirish.

$$v=\frac{-4±\sqrt{16+20}}{2}$$

16 ni 20 ga qo'shish.

$$v=\frac{-4±\sqrt{36}}{2}$$

36 ning kvadrat ildizini chiqarish.

$$v=\frac{-4±6}{2}$$

v=\frac{-4±6}{2} tenglamasini yeching, bunda ± musbat. -4 ni 6 ga qo'shish.

$$v=\frac{2}{2}$$

2 ni 2 ga bo'lish.

$$v=1$$

v=\frac{-4±6}{2} tenglamasini yeching, bunda ± manfiy. -4 dan 6 ni ayirish.

$$v=-\frac{10}{2}$$

-10 ni 2 ga bo'lish.

$$v=-5$$

Tenglama yechildi.

$$v=1 v=-5$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(v-7\right)^{2} kengaytirilishi uchun ishlating.

$$2v^{2}-10v+44=v^{2}-14v+49$$

Ikkala tarafdan v^{2} ni ayirish.

$$2v^{2}-10v+44-v^{2}=-14v+49$$

v^{2} ni olish uchun 2v^{2} va -v^{2} ni birlashtirish.

$$v^{2}-10v+44=-14v+49$$

14v ni ikki tarafga qo’shing.

$$v^{2}-10v+44+14v=49$$

4v ni olish uchun -10v va 14v ni birlashtirish.

$$v^{2}+4v+44=49$$

Ikkala tarafdan 44 ni ayirish.

$$v^{2}+4v=49-44$$

5 olish uchun 49 dan 44 ni ayirish.

$$v^{2}+4v=5$$

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.

$$v^{2}+4v+2^{2}=5+2^{2}$$

2 kvadratini chiqarish.

$$v^{2}+4v+4=5+4$$

5 ni 4 ga qo'shish.

$$v^{2}+4v+4=9$$

v^{2}+4v+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.

$$\left(v+2\right)^{2}=9$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(v+2\right)^{2}}=\sqrt{9}$$

Qisqartirish.

$$v+2=3 v+2=-3$$

Tenglamaning ikkala tarafidan 2 ni ayirish.

$$v=1 v=-5$$