12x^2+49x+44 eching

$$12 x ^ { 2 } + 49 x + 44$$

Omil

$$\left(3x+4\right)\left(4x+11\right)$$

Yechim qadamlari

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 12x^{2}+ax+bx+44 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=49 ab=12\times 44=528$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 528-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,528 2,264 3,176 4,132 6,88 8,66 11,48 12,44 16,33 22,24$$

Har bir juftlik yigʻindisini hisoblang.

$$1+528=529 2+264=266 3+176=179 4+132=136 6+88=94 8+66=74 11+48=59 12+44=56 16+33=49 22+24=46$$

Yechim – 49 yigʻindisini beruvchi juftlik.

$$a=16 b=33$$

12x^{2}+49x+44 ni \left(12x^{2}+16x\right)+\left(33x+44\right) sifatida qaytadan yozish.

$$\left(12x^{2}+16x\right)+\left(33x+44\right)$$

Birinchi guruhda 4x ni va ikkinchi guruhda 11 ni faktordan chiqaring.

$$4x\left(3x+4\right)+11\left(3x+4\right)$$

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

$$\left(3x+4\right)\left(4x+11\right)$$

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.

$$12x^{2}+49x+44=0$$

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.

$$x=\frac{-49±\sqrt{49^{2}-4\times 12\times 44}}{2\times 12}$$

49 kvadratini chiqarish.

$$x=\frac{-49±\sqrt{2401-4\times 12\times 44}}{2\times 12}$$

-4 ni 12 marotabaga ko'paytirish.

$$x=\frac{-49±\sqrt{2401-48\times 44}}{2\times 12}$$

-48 ni 44 marotabaga ko'paytirish.

$$x=\frac{-49±\sqrt{2401-2112}}{2\times 12}$$

2401 ni -2112 ga qo'shish.

$$x=\frac{-49±\sqrt{289}}{2\times 12}$$

289 ning kvadrat ildizini chiqarish.

$$x=\frac{-49±17}{2\times 12}$$

2 ni 12 marotabaga ko'paytirish.

$$x=\frac{-49±17}{24}$$

x=\frac{-49±17}{24} tenglamasini yeching, bunda ± musbat. -49 ni 17 ga qo'shish.

$$x=-\frac{32}{24}$$

\frac{-32}{24} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{-49±17}{24} tenglamasini yeching, bunda ± manfiy. -49 dan 17 ni ayirish.

$$x=-\frac{66}{24}$$

\frac{-66}{24} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{11}{4}$$

ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right) formulasi yordamida amalni hisoblang. x_{1} uchun -\frac{4}{3} ga va x_{2} uchun -\frac{11}{4} ga bo‘ling.

$$12x^{2}+49x+44=12\left(x-\left(-\frac{4}{3}\right)\right)\left(x-\left(-\frac{11}{4}\right)\right)$$

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

$$12x^{2}+49x+44=12\left(x+\frac{4}{3}\right)\left(x+\frac{11}{4}\right)$$

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

$$12x^{2}+49x+44=12\times \frac{3x+4}{3}\left(x+\frac{11}{4}\right)$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{11}{4} ni x ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$12x^{2}+49x+44=12\times \frac{3x+4}{3}\times \frac{4x+11}{4}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{3x+4}{3} ni \frac{4x+11}{4} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$12x^{2}+49x+44=12\times \frac{\left(3x+4\right)\left(4x+11\right)}{3\times 4}$$

3 ni 4 marotabaga ko'paytirish.

$$12x^{2}+49x+44=12\times \frac{\left(3x+4\right)\left(4x+11\right)}{12}$$

12 va 12 ichida eng katta umumiy 12 faktorini bekor qiling.

$$12x^{2}+49x+44=\left(3x+4\right)\left(4x+11\right)$$

Baholash

$$\left(3x+4\right)\left(4x+11\right)$$