x^2+2x=(x+2)(1-x) eching

$$x ^ { 2 } + 2 x = ( x + 2 ) ( 1 - x )$$

x uchun yechish

$$x=-2$$

$$x=\frac{1}{2}=0,5$$

Guruhlash orqali faktorlash qadamlari

x+2 ga 1-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

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

x ni ikki tarafga qo’shing.

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

3x ni olish uchun 2x va x ni birlashtirish.

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

x^{2} ni ikki tarafga qo’shing.

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

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

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

Ikkala tarafdan 2 ni ayirish.

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

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

$$a+b=3 ab=2\left(-2\right)=-4$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -4-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,4 -2,2$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+4=3 -2+2=0$$

Yechim – 3 yigʻindisini beruvchi juftlik.

$$a=-1 b=4$$

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

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

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

$$x\left(2x-1\right)+2\left(2x-1\right)$$

Distributiv funktsiyasidan foydalangan holda 2x-1 umumiy terminini chiqaring.

$$\left(2x-1\right)\left(x+2\right)$$

Tenglamani yechish uchun 2x-1=0 va x+2=0 ni yeching.

$$x=\frac{1}{2} x=-2$$

Kvadrat tenglamalaridan foydalanish qoidalari

x+2 ga 1-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

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

x ni ikki tarafga qo’shing.

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

3x ni olish uchun 2x va x ni birlashtirish.

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

x^{2} ni ikki tarafga qo’shing.

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

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

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

Ikkala tarafdan 2 ni ayirish.

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

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 -2 ni c bilan almashtiring.

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

3 kvadratini chiqarish.

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

-4 ni 2 marotabaga ko'paytirish.

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

-8 ni -2 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9+16}}{2\times 2}$$

9 ni 16 ga qo'shish.

$$x=\frac{-3±\sqrt{25}}{2\times 2}$$

25 ning kvadrat ildizini chiqarish.

$$x=\frac{-3±5}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{-3±5}{4}$$

x=\frac{-3±5}{4} tenglamasini yeching, bunda ± musbat. -3 ni 5 ga qo'shish.

$$x=\frac{2}{4}$$

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

$$x=\frac{1}{2}$$

x=\frac{-3±5}{4} tenglamasini yeching, bunda ± manfiy. -3 dan 5 ni ayirish.

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

-8 ni 4 ga bo'lish.

$$x=-2$$

Tenglama yechildi.

$$x=\frac{1}{2} x=-2$$

Kvadratni yakunlash bo'yicha qadamlar

x+2 ga 1-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

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

x ni ikki tarafga qo’shing.

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

3x ni olish uchun 2x va x ni birlashtirish.

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

x^{2} ni ikki tarafga qo’shing.

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

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

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

Ikki tarafini 2 ga bo‘ling.

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

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

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

2 ni 2 ga bo'lish.

$$x^{2}+\frac{3}{2}x=1$$

\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.

$$x^{2}+\frac{3}{2}x+\left(\frac{3}{4}\right)^{2}=1+\left(\frac{3}{4}\right)^{2}$$

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

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=1+\frac{9}{16}$$

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

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{25}{16}$$

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.

$$\left(x+\frac{3}{4}\right)^{2}=\frac{25}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{3}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Qisqartirish.

$$x+\frac{3}{4}=\frac{5}{4} x+\frac{3}{4}=-\frac{5}{4}$$

Tenglamaning ikkala tarafidan \frac{3}{4} ni ayirish.

$$x=\frac{1}{2} x=-2$$