11y^2+y=2 eching

$$11 y ^ { 2 } + y = 2$$

y uchun yechish

$$y=\frac{\sqrt{89}-1}{22}\approx 0,383362779$$

$$y=\frac{-\sqrt{89}-1}{22}\approx -0,47427187$$

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.

$$11y^{2}+y=2$$

Tenglamaning ikkala tarafidan 2 ni ayirish.

$$11y^{2}+y-2=2-2$$

O‘zidan 2 ayirilsa 0 qoladi.

$$11y^{2}+y-2=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 11 ni a, 1 ni b va -2 ni c bilan almashtiring.

$$y=\frac{-1±\sqrt{1^{2}-4\times 11\left(-2\right)}}{2\times 11}$$

1 kvadratini chiqarish.

$$y=\frac{-1±\sqrt{1-4\times 11\left(-2\right)}}{2\times 11}$$

-4 ni 11 marotabaga ko'paytirish.

$$y=\frac{-1±\sqrt{1-44\left(-2\right)}}{2\times 11}$$

-44 ni -2 marotabaga ko'paytirish.

$$y=\frac{-1±\sqrt{1+88}}{2\times 11}$$

1 ni 88 ga qo'shish.

$$y=\frac{-1±\sqrt{89}}{2\times 11}$$

2 ni 11 marotabaga ko'paytirish.

$$y=\frac{-1±\sqrt{89}}{22}$$

y=\frac{-1±\sqrt{89}}{22} tenglamasini yeching, bunda ± musbat. -1 ni \sqrt{89} ga qo'shish.

$$y=\frac{\sqrt{89}-1}{22}$$

y=\frac{-1±\sqrt{89}}{22} tenglamasini yeching, bunda ± manfiy. -1 dan \sqrt{89} ni ayirish.

$$y=\frac{-\sqrt{89}-1}{22}$$

Tenglama yechildi.

$$y=\frac{\sqrt{89}-1}{22} y=\frac{-\sqrt{89}-1}{22}$$

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.

$$11y^{2}+y=2$$

Ikki tarafini 11 ga bo‘ling.

$$\frac{11y^{2}+y}{11}=\frac{2}{11}$$

11 ga bo'lish 11 ga ko'paytirishni bekor qiladi.

$$y^{2}+\frac{1}{11}y=\frac{2}{11}$$

\frac{1}{11} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{22} olish uchun. Keyin, \frac{1}{22} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$y^{2}+\frac{1}{11}y+\left(\frac{1}{22}\right)^{2}=\frac{2}{11}+\left(\frac{1}{22}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{22} kvadratini chiqarish.

$$y^{2}+\frac{1}{11}y+\frac{1}{484}=\frac{2}{11}+\frac{1}{484}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{2}{11} ni \frac{1}{484} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$y^{2}+\frac{1}{11}y+\frac{1}{484}=\frac{89}{484}$$

y^{2}+\frac{1}{11}y+\frac{1}{484} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(y+\frac{1}{22}\right)^{2}=\frac{89}{484}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(y+\frac{1}{22}\right)^{2}}=\sqrt{\frac{89}{484}}$$

Qisqartirish.

$$y+\frac{1}{22}=\frac{\sqrt{89}}{22} y+\frac{1}{22}=-\frac{\sqrt{89}}{22}$$

Tenglamaning ikkala tarafidan \frac{1}{22} ni ayirish.

$$y=\frac{\sqrt{89}-1}{22} y=\frac{-\sqrt{89}-1}{22}$$