22p^2+51p-10 eching

$$22 p ^ { 2 } + 51 p - 10$$

Omil

$$\left(11p-2\right)\left(2p+5\right)$$

Yechim qadamlari

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 22p^{2}+ap+bp-10 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=51 ab=22\left(-10\right)=-220$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -220-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,220 -2,110 -4,55 -5,44 -10,22 -11,20$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+220=219 -2+110=108 -4+55=51 -5+44=39 -10+22=12 -11+20=9$$

Yechim – 51 yigʻindisini beruvchi juftlik.

$$a=-4 b=55$$

22p^{2}+51p-10 ni \left(22p^{2}-4p\right)+\left(55p-10\right) sifatida qaytadan yozish.

$$\left(22p^{2}-4p\right)+\left(55p-10\right)$$

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

$$2p\left(11p-2\right)+5\left(11p-2\right)$$

Distributiv funktsiyasidan foydalangan holda 11p-2 umumiy terminini chiqaring.

$$\left(11p-2\right)\left(2p+5\right)$$

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.

$$22p^{2}+51p-10=0$$

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.

$$p=\frac{-51±\sqrt{51^{2}-4\times 22\left(-10\right)}}{2\times 22}$$

51 kvadratini chiqarish.

$$p=\frac{-51±\sqrt{2601-4\times 22\left(-10\right)}}{2\times 22}$$

-4 ni 22 marotabaga ko'paytirish.

$$p=\frac{-51±\sqrt{2601-88\left(-10\right)}}{2\times 22}$$

-88 ni -10 marotabaga ko'paytirish.

$$p=\frac{-51±\sqrt{2601+880}}{2\times 22}$$

2601 ni 880 ga qo'shish.

$$p=\frac{-51±\sqrt{3481}}{2\times 22}$$

3481 ning kvadrat ildizini chiqarish.

$$p=\frac{-51±59}{2\times 22}$$

2 ni 22 marotabaga ko'paytirish.

$$p=\frac{-51±59}{44}$$

p=\frac{-51±59}{44} tenglamasini yeching, bunda ± musbat. -51 ni 59 ga qo'shish.

$$p=\frac{8}{44}$$

\frac{8}{44} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$p=\frac{2}{11}$$

p=\frac{-51±59}{44} tenglamasini yeching, bunda ± manfiy. -51 dan 59 ni ayirish.

$$p=-\frac{110}{44}$$

\frac{-110}{44} ulushini 22 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right) formulasi yordamida amalni hisoblang. x_{1} uchun \frac{2}{11} ga va x_{2} uchun -\frac{5}{2} ga bo‘ling.

$$22p^{2}+51p-10=22\left(p-\frac{2}{11}\right)\left(p-\left(-\frac{5}{2}\right)\right)$$

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

$$22p^{2}+51p-10=22\left(p-\frac{2}{11}\right)\left(p+\frac{5}{2}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{2}{11} ni p dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$22p^{2}+51p-10=22\times \frac{11p-2}{11}\left(p+\frac{5}{2}\right)$$

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

$$22p^{2}+51p-10=22\times \frac{11p-2}{11}\times \frac{2p+5}{2}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{11p-2}{11} ni \frac{2p+5}{2} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$22p^{2}+51p-10=22\times \frac{\left(11p-2\right)\left(2p+5\right)}{11\times 2}$$

11 ni 2 marotabaga ko'paytirish.

$$22p^{2}+51p-10=22\times \frac{\left(11p-2\right)\left(2p+5\right)}{22}$$

22 va 22 ichida eng katta umumiy 22 faktorini bekor qiling.

$$22p^{2}+51p-10=\left(11p-2\right)\left(2p+5\right)$$

Baholash

$$\left(11p-2\right)\left(2p+5\right)$$