left(2x-3right)^2=49 eching

$${ \left(2x-3 \right) }^{ 2 } =49$$

x uchun yechish

$$x=5$$

$$x=-2$$

Guruhlash orqali faktorlash qadamlari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(2x-3\right)^{2} kengaytirilishi uchun ishlating.

$$4x^{2}-12x+9=49$$

Ikkala tarafdan 49 ni ayirish.

$$4x^{2}-12x+9-49=0$$

-40 olish uchun 9 dan 49 ni ayirish.

$$4x^{2}-12x-40=0$$

Ikki tarafini 4 ga bo‘ling.

$$x^{2}-3x-10=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon x^{2}+ax+bx-10 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-3 ab=1\left(-10\right)=-10$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -10-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-10 2,-5$$

Har bir juftlik yigʻindisini hisoblang.

$$1-10=-9 2-5=-3$$

Yechim – -3 yigʻindisini beruvchi juftlik.

$$a=-5 b=2$$

x^{2}-3x-10 ni \left(x^{2}-5x\right)+\left(2x-10\right) sifatida qaytadan yozish.

$$\left(x^{2}-5x\right)+\left(2x-10\right)$$

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

$$x\left(x-5\right)+2\left(x-5\right)$$

Distributiv funktsiyasidan foydalangan holda x-5 umumiy terminini chiqaring.

$$\left(x-5\right)\left(x+2\right)$$

Tenglamani yechish uchun x-5=0 va x+2=0 ni yeching.

$$x=5 x=-2$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(2x-3\right)^{2} kengaytirilishi uchun ishlating.

$$4x^{2}-12x+9=49$$

Ikkala tarafdan 49 ni ayirish.

$$4x^{2}-12x+9-49=0$$

-40 olish uchun 9 dan 49 ni ayirish.

$$4x^{2}-12x-40=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -12 ni b va -40 ni c bilan almashtiring.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 4\left(-40\right)}}{2\times 4}$$

-12 kvadratini chiqarish.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 4\left(-40\right)}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-12\right)±\sqrt{144-16\left(-40\right)}}{2\times 4}$$

-16 ni -40 marotabaga ko'paytirish.

$$x=\frac{-\left(-12\right)±\sqrt{144+640}}{2\times 4}$$

144 ni 640 ga qo'shish.

$$x=\frac{-\left(-12\right)±\sqrt{784}}{2\times 4}$$

784 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-12\right)±28}{2\times 4}$$

-12 ning teskarisi 12 ga teng.

$$x=\frac{12±28}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{12±28}{8}$$

x=\frac{12±28}{8} tenglamasini yeching, bunda ± musbat. 12 ni 28 ga qo'shish.

$$x=\frac{40}{8}$$

40 ni 8 ga bo'lish.

$$x=5$$

x=\frac{12±28}{8} tenglamasini yeching, bunda ± manfiy. 12 dan 28 ni ayirish.

$$x=-\frac{16}{8}$$

-16 ni 8 ga bo'lish.

$$x=-2$$

Tenglama yechildi.

$$x=5 x=-2$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(2x-3\right)^{2} kengaytirilishi uchun ishlating.

$$4x^{2}-12x+9=49$$

Ikkala tarafdan 9 ni ayirish.

$$4x^{2}-12x=49-9$$

40 olish uchun 49 dan 9 ni ayirish.

$$4x^{2}-12x=40$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}-12x}{4}=\frac{40}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{12}{4}\right)x=\frac{40}{4}$$

-12 ni 4 ga bo'lish.

$$x^{2}-3x=\frac{40}{4}$$

40 ni 4 ga bo'lish.

$$x^{2}-3x=10$$

-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.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=10+\left(-\frac{3}{2}\right)^{2}$$

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

$$x^{2}-3x+\frac{9}{4}=10+\frac{9}{4}$$

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

$$x^{2}-3x+\frac{9}{4}=\frac{49}{4}$$

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.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{49}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Qisqartirish.

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

\frac{3}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=5 x=-2$$