8x^2-14x-15 eching

$$8 { x }^{ 2 } -14x-15$$

Omil

$$\left(2x-5\right)\left(4x+3\right)$$

Yechim qadamlari

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 8x^{2}+ax+bx-15 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-14 ab=8\left(-15\right)=-120$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -120-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

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

Har bir juftlik yigʻindisini hisoblang.

$$1-120=-119 2-60=-58 3-40=-37 4-30=-26 5-24=-19 6-20=-14 8-15=-7 10-12=-2$$

Yechim – -14 yigʻindisini beruvchi juftlik.

$$a=-20 b=6$$

8x^{2}-14x-15 ni \left(8x^{2}-20x\right)+\left(6x-15\right) sifatida qaytadan yozish.

$$\left(8x^{2}-20x\right)+\left(6x-15\right)$$

Birinchi guruhda 4x ni va ikkinchi guruhda 3 ni faktordan chiqaring.

$$4x\left(2x-5\right)+3\left(2x-5\right)$$

Distributiv funktsiyasidan foydalangan holda 2x-5 umumiy terminini chiqaring.

$$\left(2x-5\right)\left(4x+3\right)$$

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.

$$8x^{2}-14x-15=0$$

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.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 8\left(-15\right)}}{2\times 8}$$

-14 kvadratini chiqarish.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 8\left(-15\right)}}{2\times 8}$$

-4 ni 8 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196-32\left(-15\right)}}{2\times 8}$$

-32 ni -15 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196+480}}{2\times 8}$$

196 ni 480 ga qo'shish.

$$x=\frac{-\left(-14\right)±\sqrt{676}}{2\times 8}$$

676 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-14\right)±26}{2\times 8}$$

-14 ning teskarisi 14 ga teng.

$$x=\frac{14±26}{2\times 8}$$

2 ni 8 marotabaga ko'paytirish.

$$x=\frac{14±26}{16}$$

x=\frac{14±26}{16} tenglamasini yeching, bunda ± musbat. 14 ni 26 ga qo'shish.

$$x=\frac{40}{16}$$

\frac{40}{16} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{5}{2}$$

x=\frac{14±26}{16} tenglamasini yeching, bunda ± manfiy. 14 dan 26 ni ayirish.

$$x=-\frac{12}{16}$$

\frac{-12}{16} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right) formulasi yordamida amalni hisoblang. x_{1} uchun \frac{5}{2} ga va x_{2} uchun -\frac{3}{4} ga bo‘ling.

$$8x^{2}-14x-15=8\left(x-\frac{5}{2}\right)\left(x-\left(-\frac{3}{4}\right)\right)$$

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

$$8x^{2}-14x-15=8\left(x-\frac{5}{2}\right)\left(x+\frac{3}{4}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{5}{2} ni x dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$8x^{2}-14x-15=8\times \frac{2x-5}{2}\left(x+\frac{3}{4}\right)$$

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

$$8x^{2}-14x-15=8\times \frac{2x-5}{2}\times \frac{4x+3}{4}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{2x-5}{2} ni \frac{4x+3}{4} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$8x^{2}-14x-15=8\times \frac{\left(2x-5\right)\left(4x+3\right)}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$8x^{2}-14x-15=8\times \frac{\left(2x-5\right)\left(4x+3\right)}{8}$$

8 va 8 ichida eng katta umumiy 8 faktorini bekor qiling.

$$8x^{2}-14x-15=\left(2x-5\right)\left(4x+3\right)$$

Baholash

$$\left(2x-5\right)\left(4x+3\right)$$