(2x+3)(x^2-16)+(x-4)(x+40)=2(x-4)(x^2-16) eching

$$( 2 x + 3 ) ( x ^ { 2 } - 16 ) + ( x - 4 ) ( x + 40 ) = 2 ( x - 4 ) ( x ^ { 2 } - 16 )$$

x uchun yechish

$$x=-7$$

$$x=4$$

Guruhlash orqali faktorlash qadamlari

2x+3 ga x^{2}-16 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{3}-32x+3x^{2}-48+\left(x-4\right)\left(x+40\right)=2\left(x-4\right)\left(x^{2}-16\right)$$

x-4 ga x+40 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{3}-32x+3x^{2}-48+x^{2}+36x-160=2\left(x-4\right)\left(x^{2}-16\right)$$

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

$$2x^{3}-32x+4x^{2}-48+36x-160=2\left(x-4\right)\left(x^{2}-16\right)$$

4x ni olish uchun -32x va 36x ni birlashtirish.

$$2x^{3}+4x+4x^{2}-48-160=2\left(x-4\right)\left(x^{2}-16\right)$$

-208 olish uchun -48 dan 160 ni ayirish.

$$2x^{3}+4x+4x^{2}-208=2\left(x-4\right)\left(x^{2}-16\right)$$

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

$$2x^{3}+4x+4x^{2}-208=\left(2x-8\right)\left(x^{2}-16\right)$$

2x-8 ga x^{2}-16 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{3}+4x+4x^{2}-208=2x^{3}-32x-8x^{2}+128$$

Ikkala tarafdan 2x^{3} ni ayirish.

$$2x^{3}+4x+4x^{2}-208-2x^{3}=-32x-8x^{2}+128$$

0 ni olish uchun 2x^{3} va -2x^{3} ni birlashtirish.

$$4x+4x^{2}-208=-32x-8x^{2}+128$$

32x ni ikki tarafga qo’shing.

$$4x+4x^{2}-208+32x=-8x^{2}+128$$

36x ni olish uchun 4x va 32x ni birlashtirish.

$$36x+4x^{2}-208=-8x^{2}+128$$

8x^{2} ni ikki tarafga qo’shing.

$$36x+4x^{2}-208+8x^{2}=128$$

12x^{2} ni olish uchun 4x^{2} va 8x^{2} ni birlashtirish.

$$36x+12x^{2}-208=128$$

Ikkala tarafdan 128 ni ayirish.

$$36x+12x^{2}-208-128=0$$

-336 olish uchun -208 dan 128 ni ayirish.

$$36x+12x^{2}-336=0$$

Ikki tarafini 12 ga bo‘ling.

$$3x+x^{2}-28=0$$

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

$$x^{2}+3x-28=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon x^{2}+ax+bx-28 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=3 ab=1\left(-28\right)=-28$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -28-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,28 -2,14 -4,7$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+28=27 -2+14=12 -4+7=3$$

Yechim – 3 yigʻindisini beruvchi juftlik.

$$a=-4 b=7$$

x^{2}+3x-28 ni \left(x^{2}-4x\right)+\left(7x-28\right) sifatida qaytadan yozish.

$$\left(x^{2}-4x\right)+\left(7x-28\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 7 ni faktordan chiqaring.

$$x\left(x-4\right)+7\left(x-4\right)$$

Distributiv funktsiyasidan foydalangan holda x-4 umumiy terminini chiqaring.

$$\left(x-4\right)\left(x+7\right)$$

Tenglamani yechish uchun x-4=0 va x+7=0 ni yeching.

$$x=4 x=-7$$

Kvadrat tenglamalaridan foydalanish qoidalari

2x+3 ga x^{2}-16 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{3}-32x+3x^{2}-48+\left(x-4\right)\left(x+40\right)=2\left(x-4\right)\left(x^{2}-16\right)$$

x-4 ga x+40 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{3}-32x+3x^{2}-48+x^{2}+36x-160=2\left(x-4\right)\left(x^{2}-16\right)$$

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

$$2x^{3}-32x+4x^{2}-48+36x-160=2\left(x-4\right)\left(x^{2}-16\right)$$

4x ni olish uchun -32x va 36x ni birlashtirish.

$$2x^{3}+4x+4x^{2}-48-160=2\left(x-4\right)\left(x^{2}-16\right)$$

-208 olish uchun -48 dan 160 ni ayirish.

$$2x^{3}+4x+4x^{2}-208=2\left(x-4\right)\left(x^{2}-16\right)$$

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

$$2x^{3}+4x+4x^{2}-208=\left(2x-8\right)\left(x^{2}-16\right)$$

2x-8 ga x^{2}-16 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{3}+4x+4x^{2}-208=2x^{3}-32x-8x^{2}+128$$

Ikkala tarafdan 2x^{3} ni ayirish.

$$2x^{3}+4x+4x^{2}-208-2x^{3}=-32x-8x^{2}+128$$

0 ni olish uchun 2x^{3} va -2x^{3} ni birlashtirish.

$$4x+4x^{2}-208=-32x-8x^{2}+128$$

32x ni ikki tarafga qo’shing.

$$4x+4x^{2}-208+32x=-8x^{2}+128$$

36x ni olish uchun 4x va 32x ni birlashtirish.

$$36x+4x^{2}-208=-8x^{2}+128$$

8x^{2} ni ikki tarafga qo’shing.

$$36x+4x^{2}-208+8x^{2}=128$$

12x^{2} ni olish uchun 4x^{2} va 8x^{2} ni birlashtirish.

$$36x+12x^{2}-208=128$$

Ikkala tarafdan 128 ni ayirish.

$$36x+12x^{2}-208-128=0$$

-336 olish uchun -208 dan 128 ni ayirish.

$$36x+12x^{2}-336=0$$

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.

$$12x^{2}+36x-336=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 12 ni a, 36 ni b va -336 ni c bilan almashtiring.

$$x=\frac{-36±\sqrt{36^{2}-4\times 12\left(-336\right)}}{2\times 12}$$

36 kvadratini chiqarish.

$$x=\frac{-36±\sqrt{1296-4\times 12\left(-336\right)}}{2\times 12}$$

-4 ni 12 marotabaga ko'paytirish.

$$x=\frac{-36±\sqrt{1296-48\left(-336\right)}}{2\times 12}$$

-48 ni -336 marotabaga ko'paytirish.

$$x=\frac{-36±\sqrt{1296+16128}}{2\times 12}$$

1296 ni 16128 ga qo'shish.

$$x=\frac{-36±\sqrt{17424}}{2\times 12}$$

17424 ning kvadrat ildizini chiqarish.

$$x=\frac{-36±132}{2\times 12}$$

2 ni 12 marotabaga ko'paytirish.

$$x=\frac{-36±132}{24}$$

x=\frac{-36±132}{24} tenglamasini yeching, bunda ± musbat. -36 ni 132 ga qo'shish.

$$x=\frac{96}{24}$$

96 ni 24 ga bo'lish.

$$x=4$$

x=\frac{-36±132}{24} tenglamasini yeching, bunda ± manfiy. -36 dan 132 ni ayirish.

$$x=-\frac{168}{24}$$

-168 ni 24 ga bo'lish.

$$x=-7$$

Tenglama yechildi.

$$x=4 x=-7$$

Kvadratni yakunlash bo'yicha qadamlar

2x+3 ga x^{2}-16 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{3}-32x+3x^{2}-48+\left(x-4\right)\left(x+40\right)=2\left(x-4\right)\left(x^{2}-16\right)$$

x-4 ga x+40 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{3}-32x+3x^{2}-48+x^{2}+36x-160=2\left(x-4\right)\left(x^{2}-16\right)$$

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

$$2x^{3}-32x+4x^{2}-48+36x-160=2\left(x-4\right)\left(x^{2}-16\right)$$

4x ni olish uchun -32x va 36x ni birlashtirish.

$$2x^{3}+4x+4x^{2}-48-160=2\left(x-4\right)\left(x^{2}-16\right)$$

-208 olish uchun -48 dan 160 ni ayirish.

$$2x^{3}+4x+4x^{2}-208=2\left(x-4\right)\left(x^{2}-16\right)$$

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

$$2x^{3}+4x+4x^{2}-208=\left(2x-8\right)\left(x^{2}-16\right)$$

2x-8 ga x^{2}-16 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{3}+4x+4x^{2}-208=2x^{3}-32x-8x^{2}+128$$

Ikkala tarafdan 2x^{3} ni ayirish.

$$2x^{3}+4x+4x^{2}-208-2x^{3}=-32x-8x^{2}+128$$

0 ni olish uchun 2x^{3} va -2x^{3} ni birlashtirish.

$$4x+4x^{2}-208=-32x-8x^{2}+128$$

32x ni ikki tarafga qo’shing.

$$4x+4x^{2}-208+32x=-8x^{2}+128$$

36x ni olish uchun 4x va 32x ni birlashtirish.

$$36x+4x^{2}-208=-8x^{2}+128$$

8x^{2} ni ikki tarafga qo’shing.

$$36x+4x^{2}-208+8x^{2}=128$$

12x^{2} ni olish uchun 4x^{2} va 8x^{2} ni birlashtirish.

$$36x+12x^{2}-208=128$$

208 ni ikki tarafga qo’shing.

$$36x+12x^{2}=128+208$$

336 olish uchun 128 va 208'ni qo'shing.

$$36x+12x^{2}=336$$

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

$$12x^{2}+36x=336$$

Ikki tarafini 12 ga bo‘ling.

$$\frac{12x^{2}+36x}{12}=\frac{336}{12}$$

12 ga bo'lish 12 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{36}{12}x=\frac{336}{12}$$

36 ni 12 ga bo'lish.

$$x^{2}+3x=\frac{336}{12}$$

336 ni 12 ga bo'lish.

$$x^{2}+3x=28$$

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

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=28+\left(\frac{3}{2}\right)^{2}$$

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

$$x^{2}+3x+\frac{9}{4}=28+\frac{9}{4}$$

28 ni \frac{9}{4} ga qo'shish.

$$x^{2}+3x+\frac{9}{4}=\frac{121}{4}$$

x^{2}+3x+\frac{9}{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.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{121}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Qisqartirish.

$$x+\frac{3}{2}=\frac{11}{2} x+\frac{3}{2}=-\frac{11}{2}$$

Tenglamaning ikkala tarafidan \frac{3}{2} ni ayirish.

$$x=4 x=-7$$