(5x-4)^2=81 eching

$$( 5 x - 4 ) ^ { 2 } = 81$$

x uchun yechish

$$x = \frac{13}{5} = 2\frac{3}{5} = 2,6$$

$$x=-1$$

Guruhlash orqali faktorlash qadamlari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(5x-4\right)^{2} kengaytirilishi uchun ishlating.

$$25x^{2}-40x+16=81$$

Ikkala tarafdan 81 ni ayirish.

$$25x^{2}-40x+16-81=0$$

-65 olish uchun 16 dan 81 ni ayirish.

$$25x^{2}-40x-65=0$$

Ikki tarafini 5 ga bo‘ling.

$$5x^{2}-8x-13=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 5x^{2}+ax+bx-13 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-8 ab=5\left(-13\right)=-65$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -65-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-65 5,-13$$

Har bir juftlik yigʻindisini hisoblang.

$$1-65=-64 5-13=-8$$

Yechim – -8 yigʻindisini beruvchi juftlik.

$$a=-13 b=5$$

5x^{2}-8x-13 ni \left(5x^{2}-13x\right)+\left(5x-13\right) sifatida qaytadan yozish.

$$\left(5x^{2}-13x\right)+\left(5x-13\right)$$

5x^{2}-13x ichida x ni ajrating.

$$x\left(5x-13\right)+5x-13$$

Distributiv funktsiyasidan foydalangan holda 5x-13 umumiy terminini chiqaring.

$$\left(5x-13\right)\left(x+1\right)$$

Tenglamani yechish uchun 5x-13=0 va x+1=0 ni yeching.

$$x=\frac{13}{5} x=-1$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(5x-4\right)^{2} kengaytirilishi uchun ishlating.

$$25x^{2}-40x+16=81$$

Ikkala tarafdan 81 ni ayirish.

$$25x^{2}-40x+16-81=0$$

-65 olish uchun 16 dan 81 ni ayirish.

$$25x^{2}-40x-65=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 25 ni a, -40 ni b va -65 ni c bilan almashtiring.

$$x=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 25\left(-65\right)}}{2\times 25}$$

-40 kvadratini chiqarish.

$$x=\frac{-\left(-40\right)±\sqrt{1600-4\times 25\left(-65\right)}}{2\times 25}$$

-4 ni 25 marotabaga ko'paytirish.

$$x=\frac{-\left(-40\right)±\sqrt{1600-100\left(-65\right)}}{2\times 25}$$

-100 ni -65 marotabaga ko'paytirish.

$$x=\frac{-\left(-40\right)±\sqrt{1600+6500}}{2\times 25}$$

1600 ni 6500 ga qo'shish.

$$x=\frac{-\left(-40\right)±\sqrt{8100}}{2\times 25}$$

8100 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-40\right)±90}{2\times 25}$$

-40 ning teskarisi 40 ga teng.

$$x=\frac{40±90}{2\times 25}$$

2 ni 25 marotabaga ko'paytirish.

$$x=\frac{40±90}{50}$$

x=\frac{40±90}{50} tenglamasini yeching, bunda ± musbat. 40 ni 90 ga qo'shish.

$$x=\frac{130}{50}$$

\frac{130}{50} ulushini 10 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{13}{5}$$

x=\frac{40±90}{50} tenglamasini yeching, bunda ± manfiy. 40 dan 90 ni ayirish.

$$x=-\frac{50}{50}$$

-50 ni 50 ga bo'lish.

$$x=-1$$

Tenglama yechildi.

$$x=\frac{13}{5} x=-1$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(5x-4\right)^{2} kengaytirilishi uchun ishlating.

$$25x^{2}-40x+16=81$$

Ikkala tarafdan 16 ni ayirish.

$$25x^{2}-40x=81-16$$

65 olish uchun 81 dan 16 ni ayirish.

$$25x^{2}-40x=65$$

Ikki tarafini 25 ga bo‘ling.

$$\frac{25x^{2}-40x}{25}=\frac{65}{25}$$

25 ga bo'lish 25 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{40}{25}\right)x=\frac{65}{25}$$

\frac{-40}{25} ulushini 5 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{8}{5}x=\frac{65}{25}$$

\frac{65}{25} ulushini 5 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{8}{5}x=\frac{13}{5}$$

-\frac{8}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{4}{5} olish uchun. Keyin, -\frac{4}{5} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{8}{5}x+\left(-\frac{4}{5}\right)^{2}=\frac{13}{5}+\left(-\frac{4}{5}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{4}{5} kvadratini chiqarish.

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=\frac{13}{5}+\frac{16}{25}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{13}{5} ni \frac{16}{25} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=\frac{81}{25}$$

x^{2}-\frac{8}{5}x+\frac{16}{25} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{4}{5}\right)^{2}=\frac{81}{25}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{4}{5}\right)^{2}}=\sqrt{\frac{81}{25}}$$

Qisqartirish.

$$x-\frac{4}{5}=\frac{9}{5} x-\frac{4}{5}=-\frac{9}{5}$$

\frac{4}{5} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{13}{5} x=-1$$