15x^2-8x-16 eching

$$15 { x }^{ 2 } -8x-16$$

Omil

$$\left(3x-4\right)\left(5x+4\right)$$

Yechim qadamlari

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 15x^{2}+ax+bx-16 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-8 ab=15\left(-16\right)=-240$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -240-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-240 2,-120 3,-80 4,-60 5,-48 6,-40 8,-30 10,-24 12,-20 15,-16$$

Har bir juftlik yigʻindisini hisoblang.

$$1-240=-239 2-120=-118 3-80=-77 4-60=-56 5-48=-43 6-40=-34 8-30=-22 10-24=-14 12-20=-8 15-16=-1$$

Yechim – -8 yigʻindisini beruvchi juftlik.

$$a=-20 b=12$$

15x^{2}-8x-16 ni \left(15x^{2}-20x\right)+\left(12x-16\right) sifatida qaytadan yozish.

$$\left(15x^{2}-20x\right)+\left(12x-16\right)$$

Birinchi guruhda 5x ni va ikkinchi guruhda 4 ni faktordan chiqaring.

$$5x\left(3x-4\right)+4\left(3x-4\right)$$

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

$$\left(3x-4\right)\left(5x+4\right)$$

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.

$$15x^{2}-8x-16=0$$

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.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 15\left(-16\right)}}{2\times 15}$$

-8 kvadratini chiqarish.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 15\left(-16\right)}}{2\times 15}$$

-4 ni 15 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64-60\left(-16\right)}}{2\times 15}$$

-60 ni -16 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64+960}}{2\times 15}$$

64 ni 960 ga qo'shish.

$$x=\frac{-\left(-8\right)±\sqrt{1024}}{2\times 15}$$

1024 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-8\right)±32}{2\times 15}$$

-8 ning teskarisi 8 ga teng.

$$x=\frac{8±32}{2\times 15}$$

2 ni 15 marotabaga ko'paytirish.

$$x=\frac{8±32}{30}$$

x=\frac{8±32}{30} tenglamasini yeching, bunda ± musbat. 8 ni 32 ga qo'shish.

$$x=\frac{40}{30}$$

\frac{40}{30} ulushini 10 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{8±32}{30} tenglamasini yeching, bunda ± manfiy. 8 dan 32 ni ayirish.

$$x=-\frac{24}{30}$$

\frac{-24}{30} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{4}{5}$$

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{4}{5} ga bo‘ling.

$$15x^{2}-8x-16=15\left(x-\frac{4}{3}\right)\left(x-\left(-\frac{4}{5}\right)\right)$$

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

$$15x^{2}-8x-16=15\left(x-\frac{4}{3}\right)\left(x+\frac{4}{5}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{4}{3} ni x dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$15x^{2}-8x-16=15\times \frac{3x-4}{3}\left(x+\frac{4}{5}\right)$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{4}{5} ni x ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$15x^{2}-8x-16=15\times \frac{3x-4}{3}\times \frac{5x+4}{5}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{3x-4}{3} ni \frac{5x+4}{5} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$15x^{2}-8x-16=15\times \frac{\left(3x-4\right)\left(5x+4\right)}{3\times 5}$$

3 ni 5 marotabaga ko'paytirish.

$$15x^{2}-8x-16=15\times \frac{\left(3x-4\right)\left(5x+4\right)}{15}$$

15 va 15 ichida eng katta umumiy 15 faktorini bekor qiling.

$$15x^{2}-8x-16=\left(3x-4\right)\left(5x+4\right)$$

Baholash

$$\left(3x-4\right)\left(5x+4\right)$$