2y^2-y+2=0 eching

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

y uchun yechish

$$y=\frac{1+\sqrt{15}i}{4}\approx 0,25+0,968245837i$$

$$y=\frac{-\sqrt{15}i+1}{4}\approx 0,25-0,968245837i$$

Kvadrat tenglamalaridan foydalanish qoidalari

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.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -1 ni b va 2 ni c bilan almashtiring.

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

-4 ni 2 marotabaga ko'paytirish.

$$y=\frac{-\left(-1\right)±\sqrt{1-8\times 2}}{2\times 2}$$

-8 ni 2 marotabaga ko'paytirish.

$$y=\frac{-\left(-1\right)±\sqrt{1-16}}{2\times 2}$$

1 ni -16 ga qo'shish.

$$y=\frac{-\left(-1\right)±\sqrt{-15}}{2\times 2}$$

-15 ning kvadrat ildizini chiqarish.

$$y=\frac{-\left(-1\right)±\sqrt{15}i}{2\times 2}$$

-1 ning teskarisi 1 ga teng.

$$y=\frac{1±\sqrt{15}i}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$y=\frac{1±\sqrt{15}i}{4}$$

y=\frac{1±\sqrt{15}i}{4} tenglamasini yeching, bunda ± musbat. 1 ni i\sqrt{15} ga qo'shish.

$$y=\frac{1+\sqrt{15}i}{4}$$

y=\frac{1±\sqrt{15}i}{4} tenglamasini yeching, bunda ± manfiy. 1 dan i\sqrt{15} ni ayirish.

$$y=\frac{-\sqrt{15}i+1}{4}$$

Tenglama yechildi.

$$y=\frac{1+\sqrt{15}i}{4} y=\frac{-\sqrt{15}i+1}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

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

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

Tenglamaning ikkala tarafidan 2 ni ayirish.

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

O‘zidan 2 ayirilsa 0 qoladi.

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

Ikki tarafini 2 ga bo‘ling.

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

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

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

-2 ni 2 ga bo'lish.

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

-\frac{1}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{4} olish uchun. Keyin, -\frac{1}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{4} kvadratini chiqarish.

$$y^{2}-\frac{1}{2}y+\frac{1}{16}=-1+\frac{1}{16}$$

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

$$y^{2}-\frac{1}{2}y+\frac{1}{16}=-\frac{15}{16}$$

y^{2}-\frac{1}{2}y+\frac{1}{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.

$$\left(y-\frac{1}{4}\right)^{2}=-\frac{15}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(y-\frac{1}{4}\right)^{2}}=\sqrt{-\frac{15}{16}}$$

Qisqartirish.

$$y-\frac{1}{4}=\frac{\sqrt{15}i}{4} y-\frac{1}{4}=-\frac{\sqrt{15}i}{4}$$

\frac{1}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$y=\frac{1+\sqrt{15}i}{4} y=\frac{-\sqrt{15}i+1}{4}$$