4left(2x-13right)^2-9(2x-13)+2=0 eching

$$4 { \left(2x-13 \right) }^{ 2 } -9(2x-13)+2=0$$

x uchun yechish

$$x = \frac{53}{8} = 6\frac{5}{8} = 6,625$$

$$x = \frac{15}{2} = 7\frac{1}{2} = 7,5$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(2x-13\right)^{2} kengaytirilishi uchun ishlating.

$$4\left(4x^{2}-52x+169\right)-9\left(2x-13\right)+2=0$$

4 ga 4x^{2}-52x+169 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$16x^{2}-208x+676-9\left(2x-13\right)+2=0$$

-9 ga 2x-13 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$16x^{2}-208x+676-18x+117+2=0$$

-226x ni olish uchun -208x va -18x ni birlashtirish.

$$16x^{2}-226x+676+117+2=0$$

793 olish uchun 676 va 117'ni qo'shing.

$$16x^{2}-226x+793+2=0$$

795 olish uchun 793 va 2'ni qo'shing.

$$16x^{2}-226x+795=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 16 ni a, -226 ni b va 795 ni c bilan almashtiring.

$$x=\frac{-\left(-226\right)±\sqrt{\left(-226\right)^{2}-4\times 16\times 795}}{2\times 16}$$

-226 kvadratini chiqarish.

$$x=\frac{-\left(-226\right)±\sqrt{51076-4\times 16\times 795}}{2\times 16}$$

-4 ni 16 marotabaga ko'paytirish.

$$x=\frac{-\left(-226\right)±\sqrt{51076-64\times 795}}{2\times 16}$$

-64 ni 795 marotabaga ko'paytirish.

$$x=\frac{-\left(-226\right)±\sqrt{51076-50880}}{2\times 16}$$

51076 ni -50880 ga qo'shish.

$$x=\frac{-\left(-226\right)±\sqrt{196}}{2\times 16}$$

196 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-226\right)±14}{2\times 16}$$

-226 ning teskarisi 226 ga teng.

$$x=\frac{226±14}{2\times 16}$$

2 ni 16 marotabaga ko'paytirish.

$$x=\frac{226±14}{32}$$

x=\frac{226±14}{32} tenglamasini yeching, bunda ± musbat. 226 ni 14 ga qo'shish.

$$x=\frac{240}{32}$$

\frac{240}{32} ulushini 16 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{15}{2}$$

x=\frac{226±14}{32} tenglamasini yeching, bunda ± manfiy. 226 dan 14 ni ayirish.

$$x=\frac{212}{32}$$

\frac{212}{32} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{53}{8}$$

Tenglama yechildi.

$$x=\frac{15}{2} x=\frac{53}{8}$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(2x-13\right)^{2} kengaytirilishi uchun ishlating.

$$4\left(4x^{2}-52x+169\right)-9\left(2x-13\right)+2=0$$

4 ga 4x^{2}-52x+169 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$16x^{2}-208x+676-9\left(2x-13\right)+2=0$$

-9 ga 2x-13 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$16x^{2}-208x+676-18x+117+2=0$$

-226x ni olish uchun -208x va -18x ni birlashtirish.

$$16x^{2}-226x+676+117+2=0$$

793 olish uchun 676 va 117'ni qo'shing.

$$16x^{2}-226x+793+2=0$$

795 olish uchun 793 va 2'ni qo'shing.

$$16x^{2}-226x+795=0$$

Ikkala tarafdan 795 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$16x^{2}-226x=-795$$

Ikki tarafini 16 ga bo‘ling.

$$\frac{16x^{2}-226x}{16}=-\frac{795}{16}$$

16 ga bo'lish 16 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{226}{16}\right)x=-\frac{795}{16}$$

\frac{-226}{16} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{113}{8}x=-\frac{795}{16}$$

-\frac{113}{8} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{113}{16} olish uchun. Keyin, -\frac{113}{16} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{113}{8}x+\left(-\frac{113}{16}\right)^{2}=-\frac{795}{16}+\left(-\frac{113}{16}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{113}{16} kvadratini chiqarish.

$$x^{2}-\frac{113}{8}x+\frac{12769}{256}=-\frac{795}{16}+\frac{12769}{256}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{795}{16} ni \frac{12769}{256} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{113}{8}x+\frac{12769}{256}=\frac{49}{256}$$

x^{2}-\frac{113}{8}x+\frac{12769}{256} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{113}{16}\right)^{2}=\frac{49}{256}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{113}{16}\right)^{2}}=\sqrt{\frac{49}{256}}$$

Qisqartirish.

$$x-\frac{113}{16}=\frac{7}{16} x-\frac{113}{16}=-\frac{7}{16}$$

\frac{113}{16} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{15}{2} x=\frac{53}{8}$$