G(x)=-6x^2+170x+300 eching

$$G ( x ) = - 6 x ^ { 2 } + 170 x + 300$$

Omil

$$2\left(30-x\right)\left(3x+5\right)$$

Yechim qadamlari

2 omili.

$$2\left(-3x^{2}+85x+150\right)$$

Hisoblang: -3x^{2}+85x+150. Ifodani guruhlash orqali faktorlang. Avvalo, ifoda -3x^{2}+ax+bx+150 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=85 ab=-3\times 150=-450$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -450-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,450 -2,225 -3,150 -5,90 -6,75 -9,50 -10,45 -15,30 -18,25$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+450=449 -2+225=223 -3+150=147 -5+90=85 -6+75=69 -9+50=41 -10+45=35 -15+30=15 -18+25=7$$

Yechim – 85 yigʻindisini beruvchi juftlik.

$$a=90 b=-5$$

-3x^{2}+85x+150 ni \left(-3x^{2}+90x\right)+\left(-5x+150\right) sifatida qaytadan yozish.

$$\left(-3x^{2}+90x\right)+\left(-5x+150\right)$$

Birinchi guruhda 3x ni va ikkinchi guruhda 5 ni faktordan chiqaring.

$$3x\left(-x+30\right)+5\left(-x+30\right)$$

Distributiv funktsiyasidan foydalangan holda -x+30 umumiy terminini chiqaring.

$$\left(-x+30\right)\left(3x+5\right)$$

Toʻliq ajratilgan ifodani qaytadan yozing.

$$2\left(-x+30\right)\left(3x+5\right)$$

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.

$$-6x^{2}+170x+300=0$$

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.

$$x=\frac{-170±\sqrt{170^{2}-4\left(-6\right)\times 300}}{2\left(-6\right)}$$

170 kvadratini chiqarish.

$$x=\frac{-170±\sqrt{28900-4\left(-6\right)\times 300}}{2\left(-6\right)}$$

-4 ni -6 marotabaga ko'paytirish.

$$x=\frac{-170±\sqrt{28900+24\times 300}}{2\left(-6\right)}$$

24 ni 300 marotabaga ko'paytirish.

$$x=\frac{-170±\sqrt{28900+7200}}{2\left(-6\right)}$$

28900 ni 7200 ga qo'shish.

$$x=\frac{-170±\sqrt{36100}}{2\left(-6\right)}$$

36100 ning kvadrat ildizini chiqarish.

$$x=\frac{-170±190}{2\left(-6\right)}$$

2 ni -6 marotabaga ko'paytirish.

$$x=\frac{-170±190}{-12}$$

x=\frac{-170±190}{-12} tenglamasini yeching, bunda ± musbat. -170 ni 190 ga qo'shish.

$$x=\frac{20}{-12}$$

\frac{20}{-12} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{5}{3}$$

x=\frac{-170±190}{-12} tenglamasini yeching, bunda ± manfiy. -170 dan 190 ni ayirish.

$$x=-\frac{360}{-12}$$

-360 ni -12 ga bo'lish.

$$x=30$$

ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right) formulasi yordamida amalni hisoblang. x_{1} uchun -\frac{5}{3} ga va x_{2} uchun 30 ga bo‘ling.

$$-6x^{2}+170x+300=-6\left(x-\left(-\frac{5}{3}\right)\right)\left(x-30\right)$$

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

$$-6x^{2}+170x+300=-6\left(x+\frac{5}{3}\right)\left(x-30\right)$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{3} ni x ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$-6x^{2}+170x+300=-6\times \frac{-3x-5}{-3}\left(x-30\right)$$

-6 va 3 ichida eng katta umumiy 3 faktorini bekor qiling.

$$-6x^{2}+170x+300=2\left(-3x-5\right)\left(x-30\right)$$

Baholash

$$300+170x-6x^{2}$$