7x^2-74x-120 eching

$$7 { x }^{ 2 } -74x-120$$

Omil

$$\left(x-12\right)\left(7x+10\right)$$

Yechim qadamlari

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 7x^{2}+ax+bx-120 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-74 ab=7\left(-120\right)=-840$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -840-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-840 2,-420 3,-280 4,-210 5,-168 6,-140 7,-120 8,-105 10,-84 12,-70 14,-60 15,-56 20,-42 21,-40 24,-35 28,-30$$

Har bir juftlik yigʻindisini hisoblang.

$$1-840=-839 2-420=-418 3-280=-277 4-210=-206 5-168=-163 6-140=-134 7-120=-113 8-105=-97 10-84=-74 12-70=-58 14-60=-46 15-56=-41 20-42=-22 21-40=-19 24-35=-11 28-30=-2$$

Yechim – -74 yigʻindisini beruvchi juftlik.

$$a=-84 b=10$$

7x^{2}-74x-120 ni \left(7x^{2}-84x\right)+\left(10x-120\right) sifatida qaytadan yozish.

$$\left(7x^{2}-84x\right)+\left(10x-120\right)$$

Birinchi guruhda 7x ni va ikkinchi guruhda 10 ni faktordan chiqaring.

$$7x\left(x-12\right)+10\left(x-12\right)$$

Distributiv funktsiyasidan foydalangan holda x-12 umumiy terminini chiqaring.

$$\left(x-12\right)\left(7x+10\right)$$

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.

$$7x^{2}-74x-120=0$$

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.

$$x=\frac{-\left(-74\right)±\sqrt{\left(-74\right)^{2}-4\times 7\left(-120\right)}}{2\times 7}$$

-74 kvadratini chiqarish.

$$x=\frac{-\left(-74\right)±\sqrt{5476-4\times 7\left(-120\right)}}{2\times 7}$$

-4 ni 7 marotabaga ko'paytirish.

$$x=\frac{-\left(-74\right)±\sqrt{5476-28\left(-120\right)}}{2\times 7}$$

-28 ni -120 marotabaga ko'paytirish.

$$x=\frac{-\left(-74\right)±\sqrt{5476+3360}}{2\times 7}$$

5476 ni 3360 ga qo'shish.

$$x=\frac{-\left(-74\right)±\sqrt{8836}}{2\times 7}$$

8836 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-74\right)±94}{2\times 7}$$

-74 ning teskarisi 74 ga teng.

$$x=\frac{74±94}{2\times 7}$$

2 ni 7 marotabaga ko'paytirish.

$$x=\frac{74±94}{14}$$

x=\frac{74±94}{14} tenglamasini yeching, bunda ± musbat. 74 ni 94 ga qo'shish.

$$x=\frac{168}{14}$$

168 ni 14 ga bo'lish.

$$x=12$$

x=\frac{74±94}{14} tenglamasini yeching, bunda ± manfiy. 74 dan 94 ni ayirish.

$$x=-\frac{20}{14}$$

\frac{-20}{14} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{10}{7}$$

ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right) formulasi yordamida amalni hisoblang. x_{1} uchun 12 ga va x_{2} uchun -\frac{10}{7} ga bo‘ling.

$$7x^{2}-74x-120=7\left(x-12\right)\left(x-\left(-\frac{10}{7}\right)\right)$$

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

$$7x^{2}-74x-120=7\left(x-12\right)\left(x+\frac{10}{7}\right)$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{10}{7} ni x ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$7x^{2}-74x-120=7\left(x-12\right)\times \frac{7x+10}{7}$$

7 va 7 ichida eng katta umumiy 7 faktorini bekor qiling.

$$7x^{2}-74x-120=\left(x-12\right)\left(7x+10\right)$$

Baholash

$$\left(x-12\right)\left(7x+10\right)$$