5(x+2)^2=(7x+3)(x+2) eching

$$5 ( x + 2 ) ^ { 2 } = ( 7 x + 3 ) ( x + 2 )$$

x uchun yechish

$$x = \frac{7}{2} = 3\frac{1}{2} = 3,5$$

$$x=-2$$

Guruhlash orqali faktorlash qadamlari

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

$$5\left(x^{2}+4x+4\right)=\left(7x+3\right)\left(x+2\right)$$

5 ga x^{2}+4x+4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$5x^{2}+20x+20=\left(7x+3\right)\left(x+2\right)$$

7x+3 ga x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$5x^{2}+20x+20=7x^{2}+17x+6$$

Ikkala tarafdan 7x^{2} ni ayirish.

$$5x^{2}+20x+20-7x^{2}=17x+6$$

-2x^{2} ni olish uchun 5x^{2} va -7x^{2} ni birlashtirish.

$$-2x^{2}+20x+20=17x+6$$

Ikkala tarafdan 17x ni ayirish.

$$-2x^{2}+20x+20-17x=6$$

3x ni olish uchun 20x va -17x ni birlashtirish.

$$-2x^{2}+3x+20=6$$

Ikkala tarafdan 6 ni ayirish.

$$-2x^{2}+3x+20-6=0$$

14 olish uchun 20 dan 6 ni ayirish.

$$-2x^{2}+3x+14=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -2x^{2}+ax+bx+14 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=3 ab=-2\times 14=-28$$

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.

$$a=7 b=-4$$

-2x^{2}+3x+14 ni \left(-2x^{2}+7x\right)+\left(-4x+14\right) sifatida qaytadan yozish.

$$\left(-2x^{2}+7x\right)+\left(-4x+14\right)$$

Birinchi guruhda -x ni va ikkinchi guruhda -2 ni faktordan chiqaring.

$$-x\left(2x-7\right)-2\left(2x-7\right)$$

Distributiv funktsiyasidan foydalangan holda 2x-7 umumiy terminini chiqaring.

$$\left(2x-7\right)\left(-x-2\right)$$

Tenglamani yechish uchun 2x-7=0 va -x-2=0 ni yeching.

$$x=\frac{7}{2} x=-2$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$5\left(x^{2}+4x+4\right)=\left(7x+3\right)\left(x+2\right)$$

5 ga x^{2}+4x+4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$5x^{2}+20x+20=\left(7x+3\right)\left(x+2\right)$$

7x+3 ga x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$5x^{2}+20x+20=7x^{2}+17x+6$$

Ikkala tarafdan 7x^{2} ni ayirish.

$$5x^{2}+20x+20-7x^{2}=17x+6$$

-2x^{2} ni olish uchun 5x^{2} va -7x^{2} ni birlashtirish.

$$-2x^{2}+20x+20=17x+6$$

Ikkala tarafdan 17x ni ayirish.

$$-2x^{2}+20x+20-17x=6$$

3x ni olish uchun 20x va -17x ni birlashtirish.

$$-2x^{2}+3x+20=6$$

Ikkala tarafdan 6 ni ayirish.

$$-2x^{2}+3x+20-6=0$$

14 olish uchun 20 dan 6 ni ayirish.

$$-2x^{2}+3x+14=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -2 ni a, 3 ni b va 14 ni c bilan almashtiring.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-2\right)\times 14}}{2\left(-2\right)}$$

3 kvadratini chiqarish.

$$x=\frac{-3±\sqrt{9-4\left(-2\right)\times 14}}{2\left(-2\right)}$$

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9+8\times 14}}{2\left(-2\right)}$$

8 ni 14 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9+112}}{2\left(-2\right)}$$

9 ni 112 ga qo'shish.

$$x=\frac{-3±\sqrt{121}}{2\left(-2\right)}$$

121 ning kvadrat ildizini chiqarish.

$$x=\frac{-3±11}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$x=\frac{-3±11}{-4}$$

x=\frac{-3±11}{-4} tenglamasini yeching, bunda ± musbat. -3 ni 11 ga qo'shish.

$$x=\frac{8}{-4}$$

8 ni -4 ga bo'lish.

$$x=-2$$

x=\frac{-3±11}{-4} tenglamasini yeching, bunda ± manfiy. -3 dan 11 ni ayirish.

$$x=-\frac{14}{-4}$$

\frac{-14}{-4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{7}{2}$$

Tenglama yechildi.

$$x=-2 x=\frac{7}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$5\left(x^{2}+4x+4\right)=\left(7x+3\right)\left(x+2\right)$$

5 ga x^{2}+4x+4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$5x^{2}+20x+20=\left(7x+3\right)\left(x+2\right)$$

7x+3 ga x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$5x^{2}+20x+20=7x^{2}+17x+6$$

Ikkala tarafdan 7x^{2} ni ayirish.

$$5x^{2}+20x+20-7x^{2}=17x+6$$

-2x^{2} ni olish uchun 5x^{2} va -7x^{2} ni birlashtirish.

$$-2x^{2}+20x+20=17x+6$$

Ikkala tarafdan 17x ni ayirish.

$$-2x^{2}+20x+20-17x=6$$

3x ni olish uchun 20x va -17x ni birlashtirish.

$$-2x^{2}+3x+20=6$$

Ikkala tarafdan 20 ni ayirish.

$$-2x^{2}+3x=6-20$$

-14 olish uchun 6 dan 20 ni ayirish.

$$-2x^{2}+3x=-14$$

Ikki tarafini -2 ga bo‘ling.

$$\frac{-2x^{2}+3x}{-2}=-\frac{14}{-2}$$

-2 ga bo'lish -2 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{3}{-2}x=-\frac{14}{-2}$$

3 ni -2 ga bo'lish.

$$x^{2}-\frac{3}{2}x=-\frac{14}{-2}$$

-14 ni -2 ga bo'lish.

$$x^{2}-\frac{3}{2}x=7$$

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

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=7+\left(-\frac{3}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{4} kvadratini chiqarish.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=7+\frac{9}{16}$$

7 ni \frac{9}{16} ga qo'shish.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{121}{16}$$

x^{2}-\frac{3}{2}x+\frac{9}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{121}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{121}{16}}$$

Qisqartirish.

$$x-\frac{3}{4}=\frac{11}{4} x-\frac{3}{4}=-\frac{11}{4}$$

\frac{3}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{7}{2} x=-2$$