27+30x-25x^2 eching

$$27 + 30 x - 25 x ^ { 2 }$$

Omil

$$-\left(5x-9\right)\left(5x+3\right)$$

Yechim qadamlari

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$-25x^{2}+30x+27$$

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda -25x^{2}+ax+bx+27 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=30 ab=-25\times 27=-675$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -675-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,675 -3,225 -5,135 -9,75 -15,45 -25,27$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+675=674 -3+225=222 -5+135=130 -9+75=66 -15+45=30 -25+27=2$$

Yechim – 30 yigʻindisini beruvchi juftlik.

$$a=45 b=-15$$

-25x^{2}+30x+27 ni \left(-25x^{2}+45x\right)+\left(-15x+27\right) sifatida qaytadan yozish.

$$\left(-25x^{2}+45x\right)+\left(-15x+27\right)$$

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

$$-5x\left(5x-9\right)-3\left(5x-9\right)$$

Distributiv funktsiyasidan foydalangan holda 5x-9 umumiy terminini chiqaring.

$$\left(5x-9\right)\left(-5x-3\right)$$

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.

$$-25x^{2}+30x+27=0$$

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.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-25\right)\times 27}}{2\left(-25\right)}$$

30 kvadratini chiqarish.

$$x=\frac{-30±\sqrt{900-4\left(-25\right)\times 27}}{2\left(-25\right)}$$

-4 ni -25 marotabaga ko'paytirish.

$$x=\frac{-30±\sqrt{900+100\times 27}}{2\left(-25\right)}$$

100 ni 27 marotabaga ko'paytirish.

$$x=\frac{-30±\sqrt{900+2700}}{2\left(-25\right)}$$

900 ni 2700 ga qo'shish.

$$x=\frac{-30±\sqrt{3600}}{2\left(-25\right)}$$

3600 ning kvadrat ildizini chiqarish.

$$x=\frac{-30±60}{2\left(-25\right)}$$

2 ni -25 marotabaga ko'paytirish.

$$x=\frac{-30±60}{-50}$$

x=\frac{-30±60}{-50} tenglamasini yeching, bunda ± musbat. -30 ni 60 ga qo'shish.

$$x=\frac{30}{-50}$$

\frac{30}{-50} ulushini 10 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{-30±60}{-50} tenglamasini yeching, bunda ± manfiy. -30 dan 60 ni ayirish.

$$x=-\frac{90}{-50}$$

\frac{-90}{-50} ulushini 10 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{9}{5}$$

ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right) formulasi yordamida amalni hisoblang. x_{1} uchun -\frac{3}{5} ga va x_{2} uchun \frac{9}{5} ga bo‘ling.

$$-25x^{2}+30x+27=-25\left(x-\left(-\frac{3}{5}\right)\right)\left(x-\frac{9}{5}\right)$$

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

$$-25x^{2}+30x+27=-25\left(x+\frac{3}{5}\right)\left(x-\frac{9}{5}\right)$$

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

$$-25x^{2}+30x+27=-25\times \frac{-5x-3}{-5}\left(x-\frac{9}{5}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{9}{5} ni x dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$-25x^{2}+30x+27=-25\times \frac{-5x-3}{-5}\times \frac{-5x+9}{-5}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{-5x-3}{-5} ni \frac{-5x+9}{-5} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$-25x^{2}+30x+27=-25\times \frac{\left(-5x-3\right)\left(-5x+9\right)}{-5\left(-5\right)}$$

-5 ni -5 marotabaga ko'paytirish.

$$-25x^{2}+30x+27=-25\times \frac{\left(-5x-3\right)\left(-5x+9\right)}{25}$$

-25 va 25 ichida eng katta umumiy 25 faktorini bekor qiling.

$$-25x^{2}+30x+27=-\left(-5x-3\right)\left(-5x+9\right)$$

Baholash

$$27+30x-25x^{2}$$