16b^2-44b+10=0 eching

$$16 b ^ { 2 } - 44 b + 10 = 0$$

b uchun yechish

$$b=\frac{1}{4}=0,25$$

$$b = \frac{5}{2} = 2\frac{1}{2} = 2,5$$

Guruhlash orqali faktorlash qadamlari

Ikki tarafini 2 ga bo‘ling.

$$8b^{2}-22b+5=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 8b^{2}+ab+bb+5 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-22 ab=8\times 5=40$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 40-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-40 -2,-20 -4,-10 -5,-8$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-40=-41 -2-20=-22 -4-10=-14 -5-8=-13$$

Yechim – -22 yigʻindisini beruvchi juftlik.

$$a=-20 b=-2$$

8b^{2}-22b+5 ni \left(8b^{2}-20b\right)+\left(-2b+5\right) sifatida qaytadan yozish.

$$\left(8b^{2}-20b\right)+\left(-2b+5\right)$$

Birinchi guruhda 4b ni va ikkinchi guruhda -1 ni faktordan chiqaring.

$$4b\left(2b-5\right)-\left(2b-5\right)$$

Distributiv funktsiyasidan foydalangan holda 2b-5 umumiy terminini chiqaring.

$$\left(2b-5\right)\left(4b-1\right)$$

Tenglamani yechish uchun 2b-5=0 va 4b-1=0 ni yeching.

$$b=\frac{5}{2} b=\frac{1}{4}$$

Kvadrat tenglamalaridan foydalanish qoidalari

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$16b^{2}-44b+10=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 16 ni a, -44 ni b va 10 ni c bilan almashtiring.

$$b=\frac{-\left(-44\right)±\sqrt{\left(-44\right)^{2}-4\times 16\times 10}}{2\times 16}$$

-44 kvadratini chiqarish.

$$b=\frac{-\left(-44\right)±\sqrt{1936-4\times 16\times 10}}{2\times 16}$$

-4 ni 16 marotabaga ko'paytirish.

$$b=\frac{-\left(-44\right)±\sqrt{1936-64\times 10}}{2\times 16}$$

-64 ni 10 marotabaga ko'paytirish.

$$b=\frac{-\left(-44\right)±\sqrt{1936-640}}{2\times 16}$$

1936 ni -640 ga qo'shish.

$$b=\frac{-\left(-44\right)±\sqrt{1296}}{2\times 16}$$

1296 ning kvadrat ildizini chiqarish.

$$b=\frac{-\left(-44\right)±36}{2\times 16}$$

-44 ning teskarisi 44 ga teng.

$$b=\frac{44±36}{2\times 16}$$

2 ni 16 marotabaga ko'paytirish.

$$b=\frac{44±36}{32}$$

b=\frac{44±36}{32} tenglamasini yeching, bunda ± musbat. 44 ni 36 ga qo'shish.

$$b=\frac{80}{32}$$

\frac{80}{32} ulushini 16 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$b=\frac{5}{2}$$

b=\frac{44±36}{32} tenglamasini yeching, bunda ± manfiy. 44 dan 36 ni ayirish.

$$b=\frac{8}{32}$$

\frac{8}{32} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$b=\frac{1}{4}$$

Tenglama yechildi.

$$b=\frac{5}{2} b=\frac{1}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$16b^{2}-44b+10=0$$

Tenglamaning ikkala tarafidan 10 ni ayirish.

$$16b^{2}-44b+10-10=-10$$

O‘zidan 10 ayirilsa 0 qoladi.

$$16b^{2}-44b=-10$$

Ikki tarafini 16 ga bo‘ling.

$$\frac{16b^{2}-44b}{16}=-\frac{10}{16}$$

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

$$b^{2}+\left(-\frac{44}{16}\right)b=-\frac{10}{16}$$

\frac{-44}{16} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$b^{2}-\frac{11}{4}b=-\frac{10}{16}$$

\frac{-10}{16} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$b^{2}-\frac{11}{4}b=-\frac{5}{8}$$

-\frac{11}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{11}{8} olish uchun. Keyin, -\frac{11}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$b^{2}-\frac{11}{4}b+\left(-\frac{11}{8}\right)^{2}=-\frac{5}{8}+\left(-\frac{11}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{11}{8} kvadratini chiqarish.

$$b^{2}-\frac{11}{4}b+\frac{121}{64}=-\frac{5}{8}+\frac{121}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{5}{8} ni \frac{121}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$b^{2}-\frac{11}{4}b+\frac{121}{64}=\frac{81}{64}$$

b^{2}-\frac{11}{4}b+\frac{121}{64} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(b-\frac{11}{8}\right)^{2}=\frac{81}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(b-\frac{11}{8}\right)^{2}}=\sqrt{\frac{81}{64}}$$

Qisqartirish.

$$b-\frac{11}{8}=\frac{9}{8} b-\frac{11}{8}=-\frac{9}{8}$$

\frac{11}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$b=\frac{5}{2} b=\frac{1}{4}$$