3w(w+8)+w(w-4)/2-3=5-w^2 eching

$$\frac { 3 w ( w + 8 ) + w ( w - 4 ) } { 2 } - 3 = 5 - w ^ { 2 }$$

w uchun yechish

$$w=-4$$

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

Guruhlash orqali faktorlash qadamlari

Tenglamaning ikkala tarafini 2 ga ko'paytirish.

$$3w\left(w+8\right)+w\left(w-4\right)-6=10-2w^{2}$$

3w ga w+8 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3w^{2}+24w+w\left(w-4\right)-6=10-2w^{2}$$

w ga w-4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3w^{2}+24w+w^{2}-4w-6=10-2w^{2}$$

4w^{2} ni olish uchun 3w^{2} va w^{2} ni birlashtirish.

$$4w^{2}+24w-4w-6=10-2w^{2}$$

20w ni olish uchun 24w va -4w ni birlashtirish.

$$4w^{2}+20w-6=10-2w^{2}$$

Ikkala tarafdan 10 ni ayirish.

$$4w^{2}+20w-6-10=-2w^{2}$$

-16 olish uchun -6 dan 10 ni ayirish.

$$4w^{2}+20w-16=-2w^{2}$$

2w^{2} ni ikki tarafga qo’shing.

$$4w^{2}+20w-16+2w^{2}=0$$

6w^{2} ni olish uchun 4w^{2} va 2w^{2} ni birlashtirish.

$$6w^{2}+20w-16=0$$

Ikki tarafini 2 ga bo‘ling.

$$3w^{2}+10w-8=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 3w^{2}+aw+bw-8 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=10 ab=3\left(-8\right)=-24$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -24-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,24 -2,12 -3,8 -4,6$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

Yechim – 10 yigʻindisini beruvchi juftlik.

$$a=-2 b=12$$

3w^{2}+10w-8 ni \left(3w^{2}-2w\right)+\left(12w-8\right) sifatida qaytadan yozish.

$$\left(3w^{2}-2w\right)+\left(12w-8\right)$$

Birinchi guruhda w ni va ikkinchi guruhda 4 ni faktordan chiqaring.

$$w\left(3w-2\right)+4\left(3w-2\right)$$

Distributiv funktsiyasidan foydalangan holda 3w-2 umumiy terminini chiqaring.

$$\left(3w-2\right)\left(w+4\right)$$

Tenglamani yechish uchun 3w-2=0 va w+4=0 ni yeching.

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

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 2 ga ko'paytirish.

$$3w\left(w+8\right)+w\left(w-4\right)-6=10-2w^{2}$$

3w ga w+8 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3w^{2}+24w+w\left(w-4\right)-6=10-2w^{2}$$

w ga w-4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3w^{2}+24w+w^{2}-4w-6=10-2w^{2}$$

4w^{2} ni olish uchun 3w^{2} va w^{2} ni birlashtirish.

$$4w^{2}+24w-4w-6=10-2w^{2}$$

20w ni olish uchun 24w va -4w ni birlashtirish.

$$4w^{2}+20w-6=10-2w^{2}$$

Ikkala tarafdan 10 ni ayirish.

$$4w^{2}+20w-6-10=-2w^{2}$$

-16 olish uchun -6 dan 10 ni ayirish.

$$4w^{2}+20w-16=-2w^{2}$$

2w^{2} ni ikki tarafga qo’shing.

$$4w^{2}+20w-16+2w^{2}=0$$

6w^{2} ni olish uchun 4w^{2} va 2w^{2} ni birlashtirish.

$$6w^{2}+20w-16=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 6 ni a, 20 ni b va -16 ni c bilan almashtiring.

$$w=\frac{-20±\sqrt{20^{2}-4\times 6\left(-16\right)}}{2\times 6}$$

20 kvadratini chiqarish.

$$w=\frac{-20±\sqrt{400-4\times 6\left(-16\right)}}{2\times 6}$$

-4 ni 6 marotabaga ko'paytirish.

$$w=\frac{-20±\sqrt{400-24\left(-16\right)}}{2\times 6}$$

-24 ni -16 marotabaga ko'paytirish.

$$w=\frac{-20±\sqrt{400+384}}{2\times 6}$$

400 ni 384 ga qo'shish.

$$w=\frac{-20±\sqrt{784}}{2\times 6}$$

784 ning kvadrat ildizini chiqarish.

$$w=\frac{-20±28}{2\times 6}$$

2 ni 6 marotabaga ko'paytirish.

$$w=\frac{-20±28}{12}$$

w=\frac{-20±28}{12} tenglamasini yeching, bunda ± musbat. -20 ni 28 ga qo'shish.

$$w=\frac{8}{12}$$

\frac{8}{12} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$w=\frac{2}{3}$$

w=\frac{-20±28}{12} tenglamasini yeching, bunda ± manfiy. -20 dan 28 ni ayirish.

$$w=-\frac{48}{12}$$

-48 ni 12 ga bo'lish.

$$w=-4$$

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

Tenglamaning ikkala tarafini 2 ga ko'paytirish.

$$3w\left(w+8\right)+w\left(w-4\right)-6=10-2w^{2}$$

3w ga w+8 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3w^{2}+24w+w\left(w-4\right)-6=10-2w^{2}$$

w ga w-4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3w^{2}+24w+w^{2}-4w-6=10-2w^{2}$$

4w^{2} ni olish uchun 3w^{2} va w^{2} ni birlashtirish.

$$4w^{2}+24w-4w-6=10-2w^{2}$$

20w ni olish uchun 24w va -4w ni birlashtirish.

$$4w^{2}+20w-6=10-2w^{2}$$

2w^{2} ni ikki tarafga qo’shing.

$$4w^{2}+20w-6+2w^{2}=10$$

6w^{2} ni olish uchun 4w^{2} va 2w^{2} ni birlashtirish.

$$6w^{2}+20w-6=10$$

6 ni ikki tarafga qo’shing.

$$6w^{2}+20w=10+6$$

16 olish uchun 10 va 6'ni qo'shing.

$$6w^{2}+20w=16$$

Ikki tarafini 6 ga bo‘ling.

$$\frac{6w^{2}+20w}{6}=\frac{16}{6}$$

6 ga bo'lish 6 ga ko'paytirishni bekor qiladi.

$$w^{2}+\frac{20}{6}w=\frac{16}{6}$$

\frac{20}{6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$w^{2}+\frac{10}{3}w=\frac{16}{6}$$

\frac{16}{6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$w^{2}+\frac{10}{3}w=\frac{8}{3}$$

\frac{10}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{5}{3} olish uchun. Keyin, \frac{5}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$w^{2}+\frac{10}{3}w+\left(\frac{5}{3}\right)^{2}=\frac{8}{3}+\left(\frac{5}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{5}{3} kvadratini chiqarish.

$$w^{2}+\frac{10}{3}w+\frac{25}{9}=\frac{8}{3}+\frac{25}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{8}{3} ni \frac{25}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$w^{2}+\frac{10}{3}w+\frac{25}{9}=\frac{49}{9}$$

w^{2}+\frac{10}{3}w+\frac{25}{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.

$$\left(w+\frac{5}{3}\right)^{2}=\frac{49}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(w+\frac{5}{3}\right)^{2}}=\sqrt{\frac{49}{9}}$$

Qisqartirish.

$$w+\frac{5}{3}=\frac{7}{3} w+\frac{5}{3}=-\frac{7}{3}$$

Tenglamaning ikkala tarafidan \frac{5}{3} ni ayirish.

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