20x^2-9x-81 eching

$$20 x ^ { 2 } - 9 x - 81$$

Omil

$$\left(4x-9\right)\left(5x+9\right)$$

Yechim qadamlari

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 20x^{2}+ax+bx-81 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-9 ab=20\left(-81\right)=-1620$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -1620-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-1620 2,-810 3,-540 4,-405 5,-324 6,-270 9,-180 10,-162 12,-135 15,-108 18,-90 20,-81 27,-60 30,-54 36,-45$$

Har bir juftlik yigʻindisini hisoblang.

$$1-1620=-1619 2-810=-808 3-540=-537 4-405=-401 5-324=-319 6-270=-264 9-180=-171 10-162=-152 12-135=-123 15-108=-93 18-90=-72 20-81=-61 27-60=-33 30-54=-24 36-45=-9$$

Yechim – -9 yigʻindisini beruvchi juftlik.

$$a=-45 b=36$$

20x^{2}-9x-81 ni \left(20x^{2}-45x\right)+\left(36x-81\right) sifatida qaytadan yozish.

$$\left(20x^{2}-45x\right)+\left(36x-81\right)$$

Birinchi guruhda 5x ni va ikkinchi guruhda 9 ni faktordan chiqaring.

$$5x\left(4x-9\right)+9\left(4x-9\right)$$

Distributiv funktsiyasidan foydalangan holda 4x-9 umumiy terminini chiqaring.

$$\left(4x-9\right)\left(5x+9\right)$$

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.

$$20x^{2}-9x-81=0$$

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.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 20\left(-81\right)}}{2\times 20}$$

-9 kvadratini chiqarish.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 20\left(-81\right)}}{2\times 20}$$

-4 ni 20 marotabaga ko'paytirish.

$$x=\frac{-\left(-9\right)±\sqrt{81-80\left(-81\right)}}{2\times 20}$$

-80 ni -81 marotabaga ko'paytirish.

$$x=\frac{-\left(-9\right)±\sqrt{81+6480}}{2\times 20}$$

81 ni 6480 ga qo'shish.

$$x=\frac{-\left(-9\right)±\sqrt{6561}}{2\times 20}$$

6561 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-9\right)±81}{2\times 20}$$

-9 ning teskarisi 9 ga teng.

$$x=\frac{9±81}{2\times 20}$$

2 ni 20 marotabaga ko'paytirish.

$$x=\frac{9±81}{40}$$

x=\frac{9±81}{40} tenglamasini yeching, bunda ± musbat. 9 ni 81 ga qo'shish.

$$x=\frac{90}{40}$$

\frac{90}{40} ulushini 10 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{9}{4}$$

x=\frac{9±81}{40} tenglamasini yeching, bunda ± manfiy. 9 dan 81 ni ayirish.

$$x=-\frac{72}{40}$$

\frac{-72}{40} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{9}{5}$$

ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right) formulasi yordamida amalni hisoblang. x_{1} uchun \frac{9}{4} ga va x_{2} uchun -\frac{9}{5} ga bo‘ling.

$$20x^{2}-9x-81=20\left(x-\frac{9}{4}\right)\left(x-\left(-\frac{9}{5}\right)\right)$$

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

$$20x^{2}-9x-81=20\left(x-\frac{9}{4}\right)\left(x+\frac{9}{5}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{9}{4} ni x dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$20x^{2}-9x-81=20\times \frac{4x-9}{4}\left(x+\frac{9}{5}\right)$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{9}{5} ni x ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$20x^{2}-9x-81=20\times \frac{4x-9}{4}\times \frac{5x+9}{5}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{4x-9}{4} ni \frac{5x+9}{5} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$20x^{2}-9x-81=20\times \frac{\left(4x-9\right)\left(5x+9\right)}{4\times 5}$$

4 ni 5 marotabaga ko'paytirish.

$$20x^{2}-9x-81=20\times \frac{\left(4x-9\right)\left(5x+9\right)}{20}$$

20 va 20 ichida eng katta umumiy 20 faktorini bekor qiling.

$$20x^{2}-9x-81=\left(4x-9\right)\left(5x+9\right)$$

Baholash

$$\left(4x-9\right)\left(5x+9\right)$$