42m^2-89m-21 eching

$$42 m ^ { 2 } - 89 m - 21$$

Omil

$$\left(3m-7\right)\left(14m+3\right)$$

Yechim qadamlari

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 42m^{2}+am+bm-21 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-89 ab=42\left(-21\right)=-882$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -882-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-882 2,-441 3,-294 6,-147 7,-126 9,-98 14,-63 18,-49 21,-42$$

Har bir juftlik yigʻindisini hisoblang.

$$1-882=-881 2-441=-439 3-294=-291 6-147=-141 7-126=-119 9-98=-89 14-63=-49 18-49=-31 21-42=-21$$

Yechim – -89 yigʻindisini beruvchi juftlik.

$$a=-98 b=9$$

42m^{2}-89m-21 ni \left(42m^{2}-98m\right)+\left(9m-21\right) sifatida qaytadan yozish.

$$\left(42m^{2}-98m\right)+\left(9m-21\right)$$

Birinchi guruhda 14m ni va ikkinchi guruhda 3 ni faktordan chiqaring.

$$14m\left(3m-7\right)+3\left(3m-7\right)$$

Distributiv funktsiyasidan foydalangan holda 3m-7 umumiy terminini chiqaring.

$$\left(3m-7\right)\left(14m+3\right)$$

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.

$$42m^{2}-89m-21=0$$

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.

$$m=\frac{-\left(-89\right)±\sqrt{\left(-89\right)^{2}-4\times 42\left(-21\right)}}{2\times 42}$$

-89 kvadratini chiqarish.

$$m=\frac{-\left(-89\right)±\sqrt{7921-4\times 42\left(-21\right)}}{2\times 42}$$

-4 ni 42 marotabaga ko'paytirish.

$$m=\frac{-\left(-89\right)±\sqrt{7921-168\left(-21\right)}}{2\times 42}$$

-168 ni -21 marotabaga ko'paytirish.

$$m=\frac{-\left(-89\right)±\sqrt{7921+3528}}{2\times 42}$$

7921 ni 3528 ga qo'shish.

$$m=\frac{-\left(-89\right)±\sqrt{11449}}{2\times 42}$$

11449 ning kvadrat ildizini chiqarish.

$$m=\frac{-\left(-89\right)±107}{2\times 42}$$

-89 ning teskarisi 89 ga teng.

$$m=\frac{89±107}{2\times 42}$$

2 ni 42 marotabaga ko'paytirish.

$$m=\frac{89±107}{84}$$

m=\frac{89±107}{84} tenglamasini yeching, bunda ± musbat. 89 ni 107 ga qo'shish.

$$m=\frac{196}{84}$$

\frac{196}{84} ulushini 28 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$m=\frac{7}{3}$$

m=\frac{89±107}{84} tenglamasini yeching, bunda ± manfiy. 89 dan 107 ni ayirish.

$$m=-\frac{18}{84}$$

\frac{-18}{84} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$m=-\frac{3}{14}$$

ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right) formulasi yordamida amalni hisoblang. x_{1} uchun \frac{7}{3} ga va x_{2} uchun -\frac{3}{14} ga bo‘ling.

$$42m^{2}-89m-21=42\left(m-\frac{7}{3}\right)\left(m-\left(-\frac{3}{14}\right)\right)$$

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

$$42m^{2}-89m-21=42\left(m-\frac{7}{3}\right)\left(m+\frac{3}{14}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{7}{3} ni m dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$42m^{2}-89m-21=42\times \frac{3m-7}{3}\left(m+\frac{3}{14}\right)$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{14} ni m ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$42m^{2}-89m-21=42\times \frac{3m-7}{3}\times \frac{14m+3}{14}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{3m-7}{3} ni \frac{14m+3}{14} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$42m^{2}-89m-21=42\times \frac{\left(3m-7\right)\left(14m+3\right)}{3\times 14}$$

3 ni 14 marotabaga ko'paytirish.

$$42m^{2}-89m-21=42\times \frac{\left(3m-7\right)\left(14m+3\right)}{42}$$

42 va 42 ichida eng katta umumiy 42 faktorini bekor qiling.

$$42m^{2}-89m-21=\left(3m-7\right)\left(14m+3\right)$$

Baholash

$$\left(3m-7\right)\left(14m+3\right)$$