6y^2-5y-6 eching

$$6 y ^ { 2 } - 5 y - 6$$

Omil

$$\left(2y-3\right)\left(3y+2\right)$$

Yechim qadamlari

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 6y^{2}+ay+by-6 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-5 ab=6\left(-6\right)=-36$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -36-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-36 2,-18 3,-12 4,-9 6,-6$$

Har bir juftlik yigʻindisini hisoblang.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

Yechim – -5 yigʻindisini beruvchi juftlik.

$$a=-9 b=4$$

6y^{2}-5y-6 ni \left(6y^{2}-9y\right)+\left(4y-6\right) sifatida qaytadan yozish.

$$\left(6y^{2}-9y\right)+\left(4y-6\right)$$

Birinchi guruhda 3y ni va ikkinchi guruhda 2 ni faktordan chiqaring.

$$3y\left(2y-3\right)+2\left(2y-3\right)$$

Distributiv funktsiyasidan foydalangan holda 2y-3 umumiy terminini chiqaring.

$$\left(2y-3\right)\left(3y+2\right)$$

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.

$$6y^{2}-5y-6=0$$

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.

$$y=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 6\left(-6\right)}}{2\times 6}$$

-5 kvadratini chiqarish.

$$y=\frac{-\left(-5\right)±\sqrt{25-4\times 6\left(-6\right)}}{2\times 6}$$

-4 ni 6 marotabaga ko'paytirish.

$$y=\frac{-\left(-5\right)±\sqrt{25-24\left(-6\right)}}{2\times 6}$$

-24 ni -6 marotabaga ko'paytirish.

$$y=\frac{-\left(-5\right)±\sqrt{25+144}}{2\times 6}$$

25 ni 144 ga qo'shish.

$$y=\frac{-\left(-5\right)±\sqrt{169}}{2\times 6}$$

169 ning kvadrat ildizini chiqarish.

$$y=\frac{-\left(-5\right)±13}{2\times 6}$$

-5 ning teskarisi 5 ga teng.

$$y=\frac{5±13}{2\times 6}$$

2 ni 6 marotabaga ko'paytirish.

$$y=\frac{5±13}{12}$$

y=\frac{5±13}{12} tenglamasini yeching, bunda ± musbat. 5 ni 13 ga qo'shish.

$$y=\frac{18}{12}$$

\frac{18}{12} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$y=\frac{3}{2}$$

y=\frac{5±13}{12} tenglamasini yeching, bunda ± manfiy. 5 dan 13 ni ayirish.

$$y=-\frac{8}{12}$$

\frac{-8}{12} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$y=-\frac{2}{3}$$

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

$$6y^{2}-5y-6=6\left(y-\frac{3}{2}\right)\left(y-\left(-\frac{2}{3}\right)\right)$$

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

$$6y^{2}-5y-6=6\left(y-\frac{3}{2}\right)\left(y+\frac{2}{3}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{3}{2} ni y dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$6y^{2}-5y-6=6\times \frac{2y-3}{2}\left(y+\frac{2}{3}\right)$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{2}{3} ni y ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$6y^{2}-5y-6=6\times \frac{2y-3}{2}\times \frac{3y+2}{3}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{2y-3}{2} ni \frac{3y+2}{3} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$6y^{2}-5y-6=6\times \frac{\left(2y-3\right)\left(3y+2\right)}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$6y^{2}-5y-6=6\times \frac{\left(2y-3\right)\left(3y+2\right)}{6}$$

6 va 6 ichida eng katta umumiy 6 faktorini bekor qiling.

$$6y^{2}-5y-6=\left(2y-3\right)\left(3y+2\right)$$

Baholash

$$\left(2y-3\right)\left(3y+2\right)$$