8v^2+26v+15 eching

$$8 v ^ { 2 } + 26 v + 15$$

Omil

$$\left(2v+5\right)\left(4v+3\right)$$

Yechim qadamlari

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 8v^{2}+av+bv+15 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=26 ab=8\times 15=120$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 120-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,120 2,60 3,40 4,30 5,24 6,20 8,15 10,12$$

Har bir juftlik yigʻindisini hisoblang.

$$1+120=121 2+60=62 3+40=43 4+30=34 5+24=29 6+20=26 8+15=23 10+12=22$$

Yechim – 26 yigʻindisini beruvchi juftlik.

$$a=6 b=20$$

8v^{2}+26v+15 ni \left(8v^{2}+6v\right)+\left(20v+15\right) sifatida qaytadan yozish.

$$\left(8v^{2}+6v\right)+\left(20v+15\right)$$

Birinchi guruhda 2v ni va ikkinchi guruhda 5 ni faktordan chiqaring.

$$2v\left(4v+3\right)+5\left(4v+3\right)$$

Distributiv funktsiyasidan foydalangan holda 4v+3 umumiy terminini chiqaring.

$$\left(4v+3\right)\left(2v+5\right)$$

Kvadrat tenglamalaridan foydalanish qoidalari

Kvadrat koʻp tenglama bu orqali hisoblanadi: ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), bu yerda x_{1} va x_{2} ax^{2}+bx+c=0 kvadrat tenglamaning yechimlari.

$$8v^{2}+26v+15=0$$

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.

$$v=\frac{-26±\sqrt{26^{2}-4\times 8\times 15}}{2\times 8}$$

26 kvadratini chiqarish.

$$v=\frac{-26±\sqrt{676-4\times 8\times 15}}{2\times 8}$$

-4 ni 8 marotabaga ko'paytirish.

$$v=\frac{-26±\sqrt{676-32\times 15}}{2\times 8}$$

-32 ni 15 marotabaga ko'paytirish.

$$v=\frac{-26±\sqrt{676-480}}{2\times 8}$$

676 ni -480 ga qo'shish.

$$v=\frac{-26±\sqrt{196}}{2\times 8}$$

196 ning kvadrat ildizini chiqarish.

$$v=\frac{-26±14}{2\times 8}$$

2 ni 8 marotabaga ko'paytirish.

$$v=\frac{-26±14}{16}$$

v=\frac{-26±14}{16} tenglamasini yeching, bunda ± musbat. -26 ni 14 ga qo'shish.

$$v=-\frac{12}{16}$$

\frac{-12}{16} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$v=-\frac{3}{4}$$

v=\frac{-26±14}{16} tenglamasini yeching, bunda ± manfiy. -26 dan 14 ni ayirish.

$$v=-\frac{40}{16}$$

\frac{-40}{16} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$v=-\frac{5}{2}$$

ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right) formulasi yordamida amalni hisoblang. x_{1} uchun -\frac{3}{4} ga va x_{2} uchun -\frac{5}{2} ga bo‘ling.

$$8v^{2}+26v+15=8\left(v-\left(-\frac{3}{4}\right)\right)\left(v-\left(-\frac{5}{2}\right)\right)$$

p-\left(-q\right) shaklining barcha amallarigani p+q ga soddalashtiring.

$$8v^{2}+26v+15=8\left(v+\frac{3}{4}\right)\left(v+\frac{5}{2}\right)$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{4} ni v ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$8v^{2}+26v+15=8\times \frac{4v+3}{4}\left(v+\frac{5}{2}\right)$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{2} ni v ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$8v^{2}+26v+15=8\times \frac{4v+3}{4}\times \frac{2v+5}{2}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{4v+3}{4} ni \frac{2v+5}{2} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$8v^{2}+26v+15=8\times \frac{\left(4v+3\right)\left(2v+5\right)}{4\times 2}$$

4 ni 2 marotabaga ko'paytirish.

$$8v^{2}+26v+15=8\times \frac{\left(4v+3\right)\left(2v+5\right)}{8}$$

8 va 8 ichida eng katta umumiy 8 faktorini bekor qiling.

$$8v^{2}+26v+15=\left(4v+3\right)\left(2v+5\right)$$

Baholash

$$\left(2v+5\right)\left(4v+3\right)$$