(4x-3)(x^2+x+3)=(2x+1)(2x^2-3x) eching

$$( 4 x - 3 ) ( x ^ { 2 } + x + 3 ) = ( 2 x + 1 ) ( 2 x ^ { 2 } - 3 x )$$

x uchun yechish

$$x=-3$$

$$x=\frac{3}{5}=0,6$$

Guruhlash orqali faktorlash qadamlari

4x-3 ga x^{2}+x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$4x^{3}+x^{2}+9x-9=\left(2x+1\right)\left(2x^{2}-3x\right)$$

2x+1 ga 2x^{2}-3x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$4x^{3}+x^{2}+9x-9=4x^{3}-4x^{2}-3x$$

Ikkala tarafdan 4x^{3} ni ayirish.

$$4x^{3}+x^{2}+9x-9-4x^{3}=-4x^{2}-3x$$

0 ni olish uchun 4x^{3} va -4x^{3} ni birlashtirish.

$$x^{2}+9x-9=-4x^{2}-3x$$

4x^{2} ni ikki tarafga qo’shing.

$$x^{2}+9x-9+4x^{2}=-3x$$

5x^{2} ni olish uchun x^{2} va 4x^{2} ni birlashtirish.

$$5x^{2}+9x-9=-3x$$

3x ni ikki tarafga qo’shing.

$$5x^{2}+9x-9+3x=0$$

12x ni olish uchun 9x va 3x ni birlashtirish.

$$5x^{2}+12x-9=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 5x^{2}+ax+bx-9 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=12 ab=5\left(-9\right)=-45$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -45-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,45 -3,15 -5,9$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+45=44 -3+15=12 -5+9=4$$

Yechim – 12 yigʻindisini beruvchi juftlik.

$$a=-3 b=15$$

5x^{2}+12x-9 ni \left(5x^{2}-3x\right)+\left(15x-9\right) sifatida qaytadan yozish.

$$\left(5x^{2}-3x\right)+\left(15x-9\right)$$

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

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

Distributiv funktsiyasidan foydalangan holda 5x-3 umumiy terminini chiqaring.

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

Tenglamani yechish uchun 5x-3=0 va x+3=0 ni yeching.

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

Kvadrat tenglamalaridan foydalanish qoidalari

4x-3 ga x^{2}+x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$4x^{3}+x^{2}+9x-9=\left(2x+1\right)\left(2x^{2}-3x\right)$$

2x+1 ga 2x^{2}-3x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$4x^{3}+x^{2}+9x-9=4x^{3}-4x^{2}-3x$$

Ikkala tarafdan 4x^{3} ni ayirish.

$$4x^{3}+x^{2}+9x-9-4x^{3}=-4x^{2}-3x$$

0 ni olish uchun 4x^{3} va -4x^{3} ni birlashtirish.

$$x^{2}+9x-9=-4x^{2}-3x$$

4x^{2} ni ikki tarafga qo’shing.

$$x^{2}+9x-9+4x^{2}=-3x$$

5x^{2} ni olish uchun x^{2} va 4x^{2} ni birlashtirish.

$$5x^{2}+9x-9=-3x$$

3x ni ikki tarafga qo’shing.

$$5x^{2}+9x-9+3x=0$$

12x ni olish uchun 9x va 3x ni birlashtirish.

$$5x^{2}+12x-9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, 12 ni b va -9 ni c bilan almashtiring.

$$x=\frac{-12±\sqrt{12^{2}-4\times 5\left(-9\right)}}{2\times 5}$$

12 kvadratini chiqarish.

$$x=\frac{-12±\sqrt{144-4\times 5\left(-9\right)}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-12±\sqrt{144-20\left(-9\right)}}{2\times 5}$$

-20 ni -9 marotabaga ko'paytirish.

$$x=\frac{-12±\sqrt{144+180}}{2\times 5}$$

144 ni 180 ga qo'shish.

$$x=\frac{-12±\sqrt{324}}{2\times 5}$$

324 ning kvadrat ildizini chiqarish.

$$x=\frac{-12±18}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{-12±18}{10}$$

x=\frac{-12±18}{10} tenglamasini yeching, bunda ± musbat. -12 ni 18 ga qo'shish.

$$x=\frac{6}{10}$$

\frac{6}{10} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{-12±18}{10} tenglamasini yeching, bunda ± manfiy. -12 dan 18 ni ayirish.

$$x=-\frac{30}{10}$$

-30 ni 10 ga bo'lish.

$$x=-3$$

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

4x-3 ga x^{2}+x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$4x^{3}+x^{2}+9x-9=\left(2x+1\right)\left(2x^{2}-3x\right)$$

2x+1 ga 2x^{2}-3x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$4x^{3}+x^{2}+9x-9=4x^{3}-4x^{2}-3x$$

Ikkala tarafdan 4x^{3} ni ayirish.

$$4x^{3}+x^{2}+9x-9-4x^{3}=-4x^{2}-3x$$

0 ni olish uchun 4x^{3} va -4x^{3} ni birlashtirish.

$$x^{2}+9x-9=-4x^{2}-3x$$

4x^{2} ni ikki tarafga qo’shing.

$$x^{2}+9x-9+4x^{2}=-3x$$

5x^{2} ni olish uchun x^{2} va 4x^{2} ni birlashtirish.

$$5x^{2}+9x-9=-3x$$

3x ni ikki tarafga qo’shing.

$$5x^{2}+9x-9+3x=0$$

12x ni olish uchun 9x va 3x ni birlashtirish.

$$5x^{2}+12x-9=0$$

9 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$5x^{2}+12x=9$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5x^{2}+12x}{5}=\frac{9}{5}$$

5 ga bo'lish 5 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{12}{5}x=\frac{9}{5}$$

\frac{12}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{6}{5} olish uchun. Keyin, \frac{6}{5} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{12}{5}x+\left(\frac{6}{5}\right)^{2}=\frac{9}{5}+\left(\frac{6}{5}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{6}{5} kvadratini chiqarish.

$$x^{2}+\frac{12}{5}x+\frac{36}{25}=\frac{9}{5}+\frac{36}{25}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{9}{5} ni \frac{36}{25} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{12}{5}x+\frac{36}{25}=\frac{81}{25}$$

x^{2}+\frac{12}{5}x+\frac{36}{25} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{6}{5}\right)^{2}=\frac{81}{25}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{6}{5}\right)^{2}}=\sqrt{\frac{81}{25}}$$

Qisqartirish.

$$x+\frac{6}{5}=\frac{9}{5} x+\frac{6}{5}=-\frac{9}{5}$$

Tenglamaning ikkala tarafidan \frac{6}{5} ni ayirish.

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