(6x-5)^2=0 eching

$$( 6 x - 5 ) ^ { 2 } = 0$$

x uchun yechish

$$x=\frac{5}{6}\approx 0,833333333$$

Guruhlash orqali faktorlash qadamlari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(6x-5\right)^{2} kengaytirilishi uchun ishlating.

$$36x^{2}-60x+25=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 36x^{2}+ax+bx+25 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-60 ab=36\times 25=900$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 900-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-900 -2,-450 -3,-300 -4,-225 -5,-180 -6,-150 -9,-100 -10,-90 -12,-75 -15,-60 -18,-50 -20,-45 -25,-36 -30,-30$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-900=-901 -2-450=-452 -3-300=-303 -4-225=-229 -5-180=-185 -6-150=-156 -9-100=-109 -10-90=-100 -12-75=-87 -15-60=-75 -18-50=-68 -20-45=-65 -25-36=-61 -30-30=-60$$

Yechim – -60 yigʻindisini beruvchi juftlik.

$$a=-30 b=-30$$

36x^{2}-60x+25 ni \left(36x^{2}-30x\right)+\left(-30x+25\right) sifatida qaytadan yozish.

$$\left(36x^{2}-30x\right)+\left(-30x+25\right)$$

Birinchi guruhda 6x ni va ikkinchi guruhda -5 ni faktordan chiqaring.

$$6x\left(6x-5\right)-5\left(6x-5\right)$$

Distributiv funktsiyasidan foydalangan holda 6x-5 umumiy terminini chiqaring.

$$\left(6x-5\right)\left(6x-5\right)$$

Binom kvadrat sifatid qayta yozish.

$$\left(6x-5\right)^{2}$$

Tenglamani yechish uchun 6x-5=0 ni yeching.

$$x=\frac{5}{6}$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(6x-5\right)^{2} kengaytirilishi uchun ishlating.

$$36x^{2}-60x+25=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 36 ni a, -60 ni b va 25 ni c bilan almashtiring.

$$x=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 36\times 25}}{2\times 36}$$

-60 kvadratini chiqarish.

$$x=\frac{-\left(-60\right)±\sqrt{3600-4\times 36\times 25}}{2\times 36}$$

-4 ni 36 marotabaga ko'paytirish.

$$x=\frac{-\left(-60\right)±\sqrt{3600-144\times 25}}{2\times 36}$$

-144 ni 25 marotabaga ko'paytirish.

$$x=\frac{-\left(-60\right)±\sqrt{3600-3600}}{2\times 36}$$

3600 ni -3600 ga qo'shish.

$$x=\frac{-\left(-60\right)±\sqrt{0}}{2\times 36}$$

0 ning kvadrat ildizini chiqarish.

$$x=-\frac{-60}{2\times 36}$$

-60 ning teskarisi 60 ga teng.

$$x=\frac{60}{2\times 36}$$

2 ni 36 marotabaga ko'paytirish.

$$x=\frac{60}{72}$$

\frac{60}{72} ulushini 12 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{5}{6}$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(6x-5\right)^{2} kengaytirilishi uchun ishlating.

$$36x^{2}-60x+25=0$$

Ikkala tarafdan 25 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$36x^{2}-60x=-25$$

Ikki tarafini 36 ga bo‘ling.

$$\frac{36x^{2}-60x}{36}=-\frac{25}{36}$$

36 ga bo'lish 36 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{60}{36}\right)x=-\frac{25}{36}$$

\frac{-60}{36} ulushini 12 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{5}{3}x=-\frac{25}{36}$$

-\frac{5}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{5}{6} olish uchun. Keyin, -\frac{5}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{5}{3}x+\left(-\frac{5}{6}\right)^{2}=-\frac{25}{36}+\left(-\frac{5}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{5}{6} kvadratini chiqarish.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=\frac{-25+25}{36}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{25}{36} ni \frac{25}{36} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=0$$

x^{2}-\frac{5}{3}x+\frac{25}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{5}{6}\right)^{2}=0$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{5}{6}\right)^{2}}=\sqrt{0}$$

Qisqartirish.

$$x-\frac{5}{6}=0 x-\frac{5}{6}=0$$

\frac{5}{6} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{5}{6} x=\frac{5}{6}$$

Tenglama yechildi. Yechimlar bir xil.

$$x=\frac{5}{6}$$