3y^2+y-7=0 eching

$$3 y ^ { 2 } + y - 7 = 0$$

y uchun yechish

$$y = \frac{\sqrt{85} - 1}{6} \approx 1,369924076$$

$$y=\frac{-\sqrt{85}-1}{6}\approx -1,70325741$$

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.

$$3y^{2}+y-7=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, 1 ni b va -7 ni c bilan almashtiring.

$$y=\frac{-1±\sqrt{1^{2}-4\times 3\left(-7\right)}}{2\times 3}$$

1 kvadratini chiqarish.

$$y=\frac{-1±\sqrt{1-4\times 3\left(-7\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$y=\frac{-1±\sqrt{1-12\left(-7\right)}}{2\times 3}$$

-12 ni -7 marotabaga ko'paytirish.

$$y=\frac{-1±\sqrt{1+84}}{2\times 3}$$

1 ni 84 ga qo'shish.

$$y=\frac{-1±\sqrt{85}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$y=\frac{-1±\sqrt{85}}{6}$$

y=\frac{-1±\sqrt{85}}{6} tenglamasini yeching, bunda ± musbat. -1 ni \sqrt{85} ga qo'shish.

$$y=\frac{\sqrt{85}-1}{6}$$

y=\frac{-1±\sqrt{85}}{6} tenglamasini yeching, bunda ± manfiy. -1 dan \sqrt{85} ni ayirish.

$$y=\frac{-\sqrt{85}-1}{6}$$

Tenglama yechildi.

$$y=\frac{\sqrt{85}-1}{6} y=\frac{-\sqrt{85}-1}{6}$$

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.

$$3y^{2}+y-7=0$$

7 ni tenglamaning ikkala tarafiga qo'shish.

$$3y^{2}+y-7-\left(-7\right)=-\left(-7\right)$$

O‘zidan -7 ayirilsa 0 qoladi.

$$3y^{2}+y=-\left(-7\right)$$

0 dan -7 ni ayirish.

$$3y^{2}+y=7$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3y^{2}+y}{3}=\frac{7}{3}$$

3 ga bo'lish 3 ga ko'paytirishni bekor qiladi.

$$y^{2}+\frac{1}{3}y=\frac{7}{3}$$

\frac{1}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{6} olish uchun. Keyin, \frac{1}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$y^{2}+\frac{1}{3}y+\left(\frac{1}{6}\right)^{2}=\frac{7}{3}+\left(\frac{1}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{6} kvadratini chiqarish.

$$y^{2}+\frac{1}{3}y+\frac{1}{36}=\frac{7}{3}+\frac{1}{36}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{7}{3} ni \frac{1}{36} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$y^{2}+\frac{1}{3}y+\frac{1}{36}=\frac{85}{36}$$

y^{2}+\frac{1}{3}y+\frac{1}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(y+\frac{1}{6}\right)^{2}=\frac{85}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(y+\frac{1}{6}\right)^{2}}=\sqrt{\frac{85}{36}}$$

Qisqartirish.

$$y+\frac{1}{6}=\frac{\sqrt{85}}{6} y+\frac{1}{6}=-\frac{\sqrt{85}}{6}$$

Tenglamaning ikkala tarafidan \frac{1}{6} ni ayirish.

$$y=\frac{\sqrt{85}-1}{6} y=\frac{-\sqrt{85}-1}{6}$$