90m^2-137m-45 eching

$$90 m ^ { 2 } - 137 m - 45$$

Omil

$$\left(5m-9\right)\left(18m+5\right)$$

Yechim qadamlari

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 90m^{2}+am+bm-45 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-137 ab=90\left(-45\right)=-4050$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -4050-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-4050 2,-2025 3,-1350 5,-810 6,-675 9,-450 10,-405 15,-270 18,-225 25,-162 27,-150 30,-135 45,-90 50,-81 54,-75$$

Har bir juftlik yigʻindisini hisoblang.

$$1-4050=-4049 2-2025=-2023 3-1350=-1347 5-810=-805 6-675=-669 9-450=-441 10-405=-395 15-270=-255 18-225=-207 25-162=-137 27-150=-123 30-135=-105 45-90=-45 50-81=-31 54-75=-21$$

Yechim – -137 yigʻindisini beruvchi juftlik.

$$a=-162 b=25$$

90m^{2}-137m-45 ni \left(90m^{2}-162m\right)+\left(25m-45\right) sifatida qaytadan yozish.

$$\left(90m^{2}-162m\right)+\left(25m-45\right)$$

Birinchi guruhda 18m ni va ikkinchi guruhda 5 ni faktordan chiqaring.

$$18m\left(5m-9\right)+5\left(5m-9\right)$$

Distributiv funktsiyasidan foydalangan holda 5m-9 umumiy terminini chiqaring.

$$\left(5m-9\right)\left(18m+5\right)$$

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.

$$90m^{2}-137m-45=0$$

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.

$$m=\frac{-\left(-137\right)±\sqrt{\left(-137\right)^{2}-4\times 90\left(-45\right)}}{2\times 90}$$

-137 kvadratini chiqarish.

$$m=\frac{-\left(-137\right)±\sqrt{18769-4\times 90\left(-45\right)}}{2\times 90}$$

-4 ni 90 marotabaga ko'paytirish.

$$m=\frac{-\left(-137\right)±\sqrt{18769-360\left(-45\right)}}{2\times 90}$$

-360 ni -45 marotabaga ko'paytirish.

$$m=\frac{-\left(-137\right)±\sqrt{18769+16200}}{2\times 90}$$

18769 ni 16200 ga qo'shish.

$$m=\frac{-\left(-137\right)±\sqrt{34969}}{2\times 90}$$

34969 ning kvadrat ildizini chiqarish.

$$m=\frac{-\left(-137\right)±187}{2\times 90}$$

-137 ning teskarisi 137 ga teng.

$$m=\frac{137±187}{2\times 90}$$

2 ni 90 marotabaga ko'paytirish.

$$m=\frac{137±187}{180}$$

m=\frac{137±187}{180} tenglamasini yeching, bunda ± musbat. 137 ni 187 ga qo'shish.

$$m=\frac{324}{180}$$

\frac{324}{180} ulushini 36 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$m=\frac{9}{5}$$

m=\frac{137±187}{180} tenglamasini yeching, bunda ± manfiy. 137 dan 187 ni ayirish.

$$m=-\frac{50}{180}$$

\frac{-50}{180} ulushini 10 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$m=-\frac{5}{18}$$

ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right) formulasi yordamida amalni hisoblang. x_{1} uchun \frac{9}{5} ga va x_{2} uchun -\frac{5}{18} ga bo‘ling.

$$90m^{2}-137m-45=90\left(m-\frac{9}{5}\right)\left(m-\left(-\frac{5}{18}\right)\right)$$

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

$$90m^{2}-137m-45=90\left(m-\frac{9}{5}\right)\left(m+\frac{5}{18}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{9}{5} ni m dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$90m^{2}-137m-45=90\times \frac{5m-9}{5}\left(m+\frac{5}{18}\right)$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{18} ni m ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$90m^{2}-137m-45=90\times \frac{5m-9}{5}\times \frac{18m+5}{18}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{5m-9}{5} ni \frac{18m+5}{18} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$90m^{2}-137m-45=90\times \frac{\left(5m-9\right)\left(18m+5\right)}{5\times 18}$$

5 ni 18 marotabaga ko'paytirish.

$$90m^{2}-137m-45=90\times \frac{\left(5m-9\right)\left(18m+5\right)}{90}$$

90 va 90 ichida eng katta umumiy 90 faktorini bekor qiling.

$$90m^{2}-137m-45=\left(5m-9\right)\left(18m+5\right)$$

Baholash

$$\left(5m-9\right)\left(18m+5\right)$$