18k^2+25k+8=0 eching

$$18 k ^ { 2 } + 25 k + 8 = 0$$

k uchun yechish

$$k=-\frac{8}{9}\approx -0,888888889$$

$$k=-\frac{1}{2}=-0,5$$

Guruhlash orqali faktorlash qadamlari

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 18k^{2}+ak+bk+8 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=25 ab=18\times 8=144$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 144-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,144 2,72 3,48 4,36 6,24 8,18 9,16 12,12$$

Har bir juftlik yigʻindisini hisoblang.

$$1+144=145 2+72=74 3+48=51 4+36=40 6+24=30 8+18=26 9+16=25 12+12=24$$

Yechim – 25 yigʻindisini beruvchi juftlik.

$$a=9 b=16$$

18k^{2}+25k+8 ni \left(18k^{2}+9k\right)+\left(16k+8\right) sifatida qaytadan yozish.

$$\left(18k^{2}+9k\right)+\left(16k+8\right)$$

Birinchi guruhda 9k ni va ikkinchi guruhda 8 ni faktordan chiqaring.

$$9k\left(2k+1\right)+8\left(2k+1\right)$$

Distributiv funktsiyasidan foydalangan holda 2k+1 umumiy terminini chiqaring.

$$\left(2k+1\right)\left(9k+8\right)$$

Tenglamani yechish uchun 2k+1=0 va 9k+8=0 ni yeching.

$$k=-\frac{1}{2} k=-\frac{8}{9}$$

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.

$$18k^{2}+25k+8=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 18 ni a, 25 ni b va 8 ni c bilan almashtiring.

$$k=\frac{-25±\sqrt{25^{2}-4\times 18\times 8}}{2\times 18}$$

25 kvadratini chiqarish.

$$k=\frac{-25±\sqrt{625-4\times 18\times 8}}{2\times 18}$$

-4 ni 18 marotabaga ko'paytirish.

$$k=\frac{-25±\sqrt{625-72\times 8}}{2\times 18}$$

-72 ni 8 marotabaga ko'paytirish.

$$k=\frac{-25±\sqrt{625-576}}{2\times 18}$$

625 ni -576 ga qo'shish.

$$k=\frac{-25±\sqrt{49}}{2\times 18}$$

49 ning kvadrat ildizini chiqarish.

$$k=\frac{-25±7}{2\times 18}$$

2 ni 18 marotabaga ko'paytirish.

$$k=\frac{-25±7}{36}$$

k=\frac{-25±7}{36} tenglamasini yeching, bunda ± musbat. -25 ni 7 ga qo'shish.

$$k=-\frac{18}{36}$$

\frac{-18}{36} ulushini 18 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$k=-\frac{1}{2}$$

k=\frac{-25±7}{36} tenglamasini yeching, bunda ± manfiy. -25 dan 7 ni ayirish.

$$k=-\frac{32}{36}$$

\frac{-32}{36} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$k=-\frac{8}{9}$$

Tenglama yechildi.

$$k=-\frac{1}{2} k=-\frac{8}{9}$$

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.

$$18k^{2}+25k+8=0$$

Tenglamaning ikkala tarafidan 8 ni ayirish.

$$18k^{2}+25k+8-8=-8$$

O‘zidan 8 ayirilsa 0 qoladi.

$$18k^{2}+25k=-8$$

Ikki tarafini 18 ga bo‘ling.

$$\frac{18k^{2}+25k}{18}=-\frac{8}{18}$$

18 ga bo'lish 18 ga ko'paytirishni bekor qiladi.

$$k^{2}+\frac{25}{18}k=-\frac{8}{18}$$

\frac{-8}{18} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$k^{2}+\frac{25}{18}k=-\frac{4}{9}$$

\frac{25}{18} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{25}{36} olish uchun. Keyin, \frac{25}{36} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$k^{2}+\frac{25}{18}k+\left(\frac{25}{36}\right)^{2}=-\frac{4}{9}+\left(\frac{25}{36}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{25}{36} kvadratini chiqarish.

$$k^{2}+\frac{25}{18}k+\frac{625}{1296}=-\frac{4}{9}+\frac{625}{1296}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{4}{9} ni \frac{625}{1296} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$k^{2}+\frac{25}{18}k+\frac{625}{1296}=\frac{49}{1296}$$

k^{2}+\frac{25}{18}k+\frac{625}{1296} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(k+\frac{25}{36}\right)^{2}=\frac{49}{1296}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(k+\frac{25}{36}\right)^{2}}=\sqrt{\frac{49}{1296}}$$

Qisqartirish.

$$k+\frac{25}{36}=\frac{7}{36} k+\frac{25}{36}=-\frac{7}{36}$$

Tenglamaning ikkala tarafidan \frac{25}{36} ni ayirish.

$$k=-\frac{1}{2} k=-\frac{8}{9}$$