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

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

x uchun yechish

$$x = -\frac{7}{3} = -2\frac{1}{3} \approx -2,333333333$$

$$x=-3$$

Guruhlash orqali faktorlash qadamlari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2x+5\right)^{2} kengaytirilishi uchun ishlating.

$$4x^{2}+20x+25=\left(x+2\right)^{2}$$

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

$$4x^{2}+20x+25=x^{2}+4x+4$$

Ikkala tarafdan x^{2} ni ayirish.

$$4x^{2}+20x+25-x^{2}=4x+4$$

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

$$3x^{2}+20x+25=4x+4$$

Ikkala tarafdan 4x ni ayirish.

$$3x^{2}+20x+25-4x=4$$

16x ni olish uchun 20x va -4x ni birlashtirish.

$$3x^{2}+16x+25=4$$

Ikkala tarafdan 4 ni ayirish.

$$3x^{2}+16x+25-4=0$$

21 olish uchun 25 dan 4 ni ayirish.

$$3x^{2}+16x+21=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 3x^{2}+ax+bx+21 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=16 ab=3\times 21=63$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 63-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,63 3,21 7,9$$

Har bir juftlik yigʻindisini hisoblang.

$$1+63=64 3+21=24 7+9=16$$

Yechim – 16 yigʻindisini beruvchi juftlik.

$$a=7 b=9$$

3x^{2}+16x+21 ni \left(3x^{2}+7x\right)+\left(9x+21\right) sifatida qaytadan yozish.

$$\left(3x^{2}+7x\right)+\left(9x+21\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 3 ni faktordan chiqaring.

$$x\left(3x+7\right)+3\left(3x+7\right)$$

Distributiv funktsiyasidan foydalangan holda 3x+7 umumiy terminini chiqaring.

$$\left(3x+7\right)\left(x+3\right)$$

Tenglamani yechish uchun 3x+7=0 va x+3=0 ni yeching.

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

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2x+5\right)^{2} kengaytirilishi uchun ishlating.

$$4x^{2}+20x+25=\left(x+2\right)^{2}$$

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

$$4x^{2}+20x+25=x^{2}+4x+4$$

Ikkala tarafdan x^{2} ni ayirish.

$$4x^{2}+20x+25-x^{2}=4x+4$$

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

$$3x^{2}+20x+25=4x+4$$

Ikkala tarafdan 4x ni ayirish.

$$3x^{2}+20x+25-4x=4$$

16x ni olish uchun 20x va -4x ni birlashtirish.

$$3x^{2}+16x+25=4$$

Ikkala tarafdan 4 ni ayirish.

$$3x^{2}+16x+25-4=0$$

21 olish uchun 25 dan 4 ni ayirish.

$$3x^{2}+16x+21=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, 16 ni b va 21 ni c bilan almashtiring.

$$x=\frac{-16±\sqrt{16^{2}-4\times 3\times 21}}{2\times 3}$$

16 kvadratini chiqarish.

$$x=\frac{-16±\sqrt{256-4\times 3\times 21}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-16±\sqrt{256-12\times 21}}{2\times 3}$$

-12 ni 21 marotabaga ko'paytirish.

$$x=\frac{-16±\sqrt{256-252}}{2\times 3}$$

256 ni -252 ga qo'shish.

$$x=\frac{-16±\sqrt{4}}{2\times 3}$$

4 ning kvadrat ildizini chiqarish.

$$x=\frac{-16±2}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{-16±2}{6}$$

x=\frac{-16±2}{6} tenglamasini yeching, bunda ± musbat. -16 ni 2 ga qo'shish.

$$x=-\frac{14}{6}$$

\frac{-14}{6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{-16±2}{6} tenglamasini yeching, bunda ± manfiy. -16 dan 2 ni ayirish.

$$x=-\frac{18}{6}$$

-18 ni 6 ga bo'lish.

$$x=-3$$

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2x+5\right)^{2} kengaytirilishi uchun ishlating.

$$4x^{2}+20x+25=\left(x+2\right)^{2}$$

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

$$4x^{2}+20x+25=x^{2}+4x+4$$

Ikkala tarafdan x^{2} ni ayirish.

$$4x^{2}+20x+25-x^{2}=4x+4$$

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

$$3x^{2}+20x+25=4x+4$$

Ikkala tarafdan 4x ni ayirish.

$$3x^{2}+20x+25-4x=4$$

16x ni olish uchun 20x va -4x ni birlashtirish.

$$3x^{2}+16x+25=4$$

Ikkala tarafdan 25 ni ayirish.

$$3x^{2}+16x=4-25$$

-21 olish uchun 4 dan 25 ni ayirish.

$$3x^{2}+16x=-21$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}+16x}{3}=-\frac{21}{3}$$

3 ga bo'lish 3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{16}{3}x=-\frac{21}{3}$$

-21 ni 3 ga bo'lish.

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

\frac{16}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{8}{3} olish uchun. Keyin, \frac{8}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{16}{3}x+\left(\frac{8}{3}\right)^{2}=-7+\left(\frac{8}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{8}{3} kvadratini chiqarish.

$$x^{2}+\frac{16}{3}x+\frac{64}{9}=-7+\frac{64}{9}$$

-7 ni \frac{64}{9} ga qo'shish.

$$x^{2}+\frac{16}{3}x+\frac{64}{9}=\frac{1}{9}$$

x^{2}+\frac{16}{3}x+\frac{64}{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.

$$\left(x+\frac{8}{3}\right)^{2}=\frac{1}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{8}{3}\right)^{2}}=\sqrt{\frac{1}{9}}$$

Qisqartirish.

$$x+\frac{8}{3}=\frac{1}{3} x+\frac{8}{3}=-\frac{1}{3}$$

Tenglamaning ikkala tarafidan \frac{8}{3} ni ayirish.

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