(x+3)^2/5+1-(3x-1)^2/5=x(2x-3)/2 eching

$$\frac { ( x + 3 ) ^ { 2 } } { 5 } + 1 - \frac { ( 3 x - 1 ) ^ { 2 } } { 5 } = \frac { x ( 2 x - 3 ) } { 2 }$$

x uchun yechish

$$x=2$$

$$x=-\frac{1}{2}=-0,5$$

Guruhlash orqali faktorlash qadamlari

Tenglamaning ikkala tarafini 10 ga, 5,2 ning eng kichik karralisiga ko‘paytiring.

$$2\left(x+3\right)^{2}+10-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+3\right)^{2} kengaytirilishi uchun ishlating.

$$2\left(x^{2}+6x+9\right)+10-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

2 ga x^{2}+6x+9 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}+12x+18+10-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

28 olish uchun 18 va 10'ni qo'shing.

$$2x^{2}+12x+28-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(3x-1\right)^{2} kengaytirilishi uchun ishlating.

$$2x^{2}+12x+28-2\left(9x^{2}-6x+1\right)=5x\left(2x-3\right)$$

-2 ga 9x^{2}-6x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}+12x+28-18x^{2}+12x-2=5x\left(2x-3\right)$$

-16x^{2} ni olish uchun 2x^{2} va -18x^{2} ni birlashtirish.

$$-16x^{2}+12x+28+12x-2=5x\left(2x-3\right)$$

24x ni olish uchun 12x va 12x ni birlashtirish.

$$-16x^{2}+24x+28-2=5x\left(2x-3\right)$$

26 olish uchun 28 dan 2 ni ayirish.

$$-16x^{2}+24x+26=5x\left(2x-3\right)$$

5x ga 2x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-16x^{2}+24x+26=10x^{2}-15x$$

Ikkala tarafdan 10x^{2} ni ayirish.

$$-16x^{2}+24x+26-10x^{2}=-15x$$

-26x^{2} ni olish uchun -16x^{2} va -10x^{2} ni birlashtirish.

$$-26x^{2}+24x+26=-15x$$

15x ni ikki tarafga qo’shing.

$$-26x^{2}+24x+26+15x=0$$

39x ni olish uchun 24x va 15x ni birlashtirish.

$$-26x^{2}+39x+26=0$$

Ikki tarafini 13 ga bo‘ling.

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

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

$$a+b=3 ab=-2\times 2=-4$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -4-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,4 -2,2$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+4=3 -2+2=0$$

Yechim – 3 yigʻindisini beruvchi juftlik.

$$a=4 b=-1$$

-2x^{2}+3x+2 ni \left(-2x^{2}+4x\right)+\left(-x+2\right) sifatida qaytadan yozish.

$$\left(-2x^{2}+4x\right)+\left(-x+2\right)$$

-2x^{2}+4x ichida 2x ni ajrating.

$$2x\left(-x+2\right)-x+2$$

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

$$\left(-x+2\right)\left(2x+1\right)$$

Tenglamani yechish uchun -x+2=0 va 2x+1=0 ni yeching.

$$x=2 x=-\frac{1}{2}$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 10 ga, 5,2 ning eng kichik karralisiga ko‘paytiring.

$$2\left(x+3\right)^{2}+10-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+3\right)^{2} kengaytirilishi uchun ishlating.

$$2\left(x^{2}+6x+9\right)+10-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

2 ga x^{2}+6x+9 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}+12x+18+10-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

28 olish uchun 18 va 10'ni qo'shing.

$$2x^{2}+12x+28-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(3x-1\right)^{2} kengaytirilishi uchun ishlating.

$$2x^{2}+12x+28-2\left(9x^{2}-6x+1\right)=5x\left(2x-3\right)$$

-2 ga 9x^{2}-6x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}+12x+28-18x^{2}+12x-2=5x\left(2x-3\right)$$

-16x^{2} ni olish uchun 2x^{2} va -18x^{2} ni birlashtirish.

$$-16x^{2}+12x+28+12x-2=5x\left(2x-3\right)$$

24x ni olish uchun 12x va 12x ni birlashtirish.

$$-16x^{2}+24x+28-2=5x\left(2x-3\right)$$

26 olish uchun 28 dan 2 ni ayirish.

$$-16x^{2}+24x+26=5x\left(2x-3\right)$$

5x ga 2x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-16x^{2}+24x+26=10x^{2}-15x$$

Ikkala tarafdan 10x^{2} ni ayirish.

$$-16x^{2}+24x+26-10x^{2}=-15x$$

-26x^{2} ni olish uchun -16x^{2} va -10x^{2} ni birlashtirish.

$$-26x^{2}+24x+26=-15x$$

15x ni ikki tarafga qo’shing.

$$-26x^{2}+24x+26+15x=0$$

39x ni olish uchun 24x va 15x ni birlashtirish.

$$-26x^{2}+39x+26=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -26 ni a, 39 ni b va 26 ni c bilan almashtiring.

$$x=\frac{-39±\sqrt{39^{2}-4\left(-26\right)\times 26}}{2\left(-26\right)}$$

39 kvadratini chiqarish.

$$x=\frac{-39±\sqrt{1521-4\left(-26\right)\times 26}}{2\left(-26\right)}$$

-4 ni -26 marotabaga ko'paytirish.

$$x=\frac{-39±\sqrt{1521+104\times 26}}{2\left(-26\right)}$$

104 ni 26 marotabaga ko'paytirish.

$$x=\frac{-39±\sqrt{1521+2704}}{2\left(-26\right)}$$

1521 ni 2704 ga qo'shish.

$$x=\frac{-39±\sqrt{4225}}{2\left(-26\right)}$$

4225 ning kvadrat ildizini chiqarish.

$$x=\frac{-39±65}{2\left(-26\right)}$$

2 ni -26 marotabaga ko'paytirish.

$$x=\frac{-39±65}{-52}$$

x=\frac{-39±65}{-52} tenglamasini yeching, bunda ± musbat. -39 ni 65 ga qo'shish.

$$x=\frac{26}{-52}$$

\frac{26}{-52} ulushini 26 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{1}{2}$$

x=\frac{-39±65}{-52} tenglamasini yeching, bunda ± manfiy. -39 dan 65 ni ayirish.

$$x=-\frac{104}{-52}$$

-104 ni -52 ga bo'lish.

$$x=2$$

Tenglama yechildi.

$$x=-\frac{1}{2} x=2$$

Kvadratni yakunlash bo'yicha qadamlar

Tenglamaning ikkala tarafini 10 ga, 5,2 ning eng kichik karralisiga ko‘paytiring.

$$2\left(x+3\right)^{2}+10-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+3\right)^{2} kengaytirilishi uchun ishlating.

$$2\left(x^{2}+6x+9\right)+10-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

2 ga x^{2}+6x+9 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}+12x+18+10-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

28 olish uchun 18 va 10'ni qo'shing.

$$2x^{2}+12x+28-2\left(3x-1\right)^{2}=5x\left(2x-3\right)$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(3x-1\right)^{2} kengaytirilishi uchun ishlating.

$$2x^{2}+12x+28-2\left(9x^{2}-6x+1\right)=5x\left(2x-3\right)$$

-2 ga 9x^{2}-6x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}+12x+28-18x^{2}+12x-2=5x\left(2x-3\right)$$

-16x^{2} ni olish uchun 2x^{2} va -18x^{2} ni birlashtirish.

$$-16x^{2}+12x+28+12x-2=5x\left(2x-3\right)$$

24x ni olish uchun 12x va 12x ni birlashtirish.

$$-16x^{2}+24x+28-2=5x\left(2x-3\right)$$

26 olish uchun 28 dan 2 ni ayirish.

$$-16x^{2}+24x+26=5x\left(2x-3\right)$$

5x ga 2x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-16x^{2}+24x+26=10x^{2}-15x$$

Ikkala tarafdan 10x^{2} ni ayirish.

$$-16x^{2}+24x+26-10x^{2}=-15x$$

-26x^{2} ni olish uchun -16x^{2} va -10x^{2} ni birlashtirish.

$$-26x^{2}+24x+26=-15x$$

15x ni ikki tarafga qo’shing.

$$-26x^{2}+24x+26+15x=0$$

39x ni olish uchun 24x va 15x ni birlashtirish.

$$-26x^{2}+39x+26=0$$

Ikkala tarafdan 26 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$-26x^{2}+39x=-26$$

Ikki tarafini -26 ga bo‘ling.

$$\frac{-26x^{2}+39x}{-26}=-\frac{26}{-26}$$

-26 ga bo'lish -26 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{39}{-26}x=-\frac{26}{-26}$$

\frac{39}{-26} ulushini 13 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{3}{2}x=-\frac{26}{-26}$$

-26 ni -26 ga bo'lish.

$$x^{2}-\frac{3}{2}x=1$$

-\frac{3}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{4} olish uchun. Keyin, -\frac{3}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{4} kvadratini chiqarish.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=1+\frac{9}{16}$$

1 ni \frac{9}{16} ga qo'shish.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{25}{16}$$

x^{2}-\frac{3}{2}x+\frac{9}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{25}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Qisqartirish.

$$x-\frac{3}{4}=\frac{5}{4} x-\frac{3}{4}=-\frac{5}{4}$$

\frac{3}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=2 x=-\frac{1}{2}$$