2+y(1-3y)=y(y-3) eching

$$2 + y ( 1 - 3 y ) = y ( y - 3 )$$

y uchun yechish

$$y = \frac{\sqrt{3} + 1}{2} \approx 1,366025404$$

$$y=\frac{1-\sqrt{3}}{2}\approx -0,366025404$$

Kvadrat tenglamalaridan foydalanish qoidalari

y ga 1-3y ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2+y-3y^{2}=y\left(y-3\right)$$

y ga y-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2+y-3y^{2}=y^{2}-3y$$

Ikkala tarafdan y^{2} ni ayirish.

$$2+y-3y^{2}-y^{2}=-3y$$

-4y^{2} ni olish uchun -3y^{2} va -y^{2} ni birlashtirish.

$$2+y-4y^{2}=-3y$$

3y ni ikki tarafga qo’shing.

$$2+y-4y^{2}+3y=0$$

4y ni olish uchun y va 3y ni birlashtirish.

$$2+4y-4y^{2}=0$$

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.

$$-4y^{2}+4y+2=0$$

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

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

4 kvadratini chiqarish.

$$y=\frac{-4±\sqrt{16-4\left(-4\right)\times 2}}{2\left(-4\right)}$$

-4 ni -4 marotabaga ko'paytirish.

$$y=\frac{-4±\sqrt{16+16\times 2}}{2\left(-4\right)}$$

16 ni 2 marotabaga ko'paytirish.

$$y=\frac{-4±\sqrt{16+32}}{2\left(-4\right)}$$

16 ni 32 ga qo'shish.

$$y=\frac{-4±\sqrt{48}}{2\left(-4\right)}$$

48 ning kvadrat ildizini chiqarish.

$$y=\frac{-4±4\sqrt{3}}{2\left(-4\right)}$$

2 ni -4 marotabaga ko'paytirish.

$$y=\frac{-4±4\sqrt{3}}{-8}$$

y=\frac{-4±4\sqrt{3}}{-8} tenglamasini yeching, bunda ± musbat. -4 ni 4\sqrt{3} ga qo'shish.

$$y=\frac{4\sqrt{3}-4}{-8}$$

-4+4\sqrt{3} ni -8 ga bo'lish.

$$y=\frac{1-\sqrt{3}}{2}$$

y=\frac{-4±4\sqrt{3}}{-8} tenglamasini yeching, bunda ± manfiy. -4 dan 4\sqrt{3} ni ayirish.

$$y=\frac{-4\sqrt{3}-4}{-8}$$

-4-4\sqrt{3} ni -8 ga bo'lish.

$$y=\frac{\sqrt{3}+1}{2}$$

Tenglama yechildi.

$$y=\frac{1-\sqrt{3}}{2} y=\frac{\sqrt{3}+1}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

y ga 1-3y ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2+y-3y^{2}=y\left(y-3\right)$$

y ga y-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2+y-3y^{2}=y^{2}-3y$$

Ikkala tarafdan y^{2} ni ayirish.

$$2+y-3y^{2}-y^{2}=-3y$$

-4y^{2} ni olish uchun -3y^{2} va -y^{2} ni birlashtirish.

$$2+y-4y^{2}=-3y$$

3y ni ikki tarafga qo’shing.

$$2+y-4y^{2}+3y=0$$

4y ni olish uchun y va 3y ni birlashtirish.

$$2+4y-4y^{2}=0$$

Ikkala tarafdan 2 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$4y-4y^{2}=-2$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-4y^{2}+4y=-2$$

Ikki tarafini -4 ga bo‘ling.

$$\frac{-4y^{2}+4y}{-4}=-\frac{2}{-4}$$

-4 ga bo'lish -4 ga ko'paytirishni bekor qiladi.

$$y^{2}+\frac{4}{-4}y=-\frac{2}{-4}$$

4 ni -4 ga bo'lish.

$$y^{2}-y=-\frac{2}{-4}$$

\frac{-2}{-4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$y^{2}-y=\frac{1}{2}$$

-1 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{2} olish uchun. Keyin, -\frac{1}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$y^{2}-y+\left(-\frac{1}{2}\right)^{2}=\frac{1}{2}+\left(-\frac{1}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{2} kvadratini chiqarish.

$$y^{2}-y+\frac{1}{4}=\frac{1}{2}+\frac{1}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{2} ni \frac{1}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$y^{2}-y+\frac{1}{4}=\frac{3}{4}$$

y^{2}-y+\frac{1}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$y-\frac{1}{2}=\frac{\sqrt{3}}{2} y-\frac{1}{2}=-\frac{\sqrt{3}}{2}$$

\frac{1}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$y=\frac{\sqrt{3}+1}{2} y=\frac{1-\sqrt{3}}{2}$$