u^2-2/3u=5/4 eching

$$u ^ { 2 } - \frac { 2 } { 3 } u = \frac { 5 } { 4 }$$

u uchun yechish

$$u=-\frac{5}{6}\approx -0,833333333$$

$$u = \frac{3}{2} = 1\frac{1}{2} = 1,5$$

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.

$$u^{2}-\frac{2}{3}u=\frac{5}{4}$$

Tenglamaning ikkala tarafidan \frac{5}{4} ni ayirish.

$$u^{2}-\frac{2}{3}u-\frac{5}{4}=\frac{5}{4}-\frac{5}{4}$$

O‘zidan \frac{5}{4} ayirilsa 0 qoladi.

$$u^{2}-\frac{2}{3}u-\frac{5}{4}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -\frac{2}{3} ni b va -\frac{5}{4} ni c bilan almashtiring.

$$u=\frac{-\left(-\frac{2}{3}\right)±\sqrt{\left(-\frac{2}{3}\right)^{2}-4\left(-\frac{5}{4}\right)}}{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{2}{3} kvadratini chiqarish.

$$u=\frac{-\left(-\frac{2}{3}\right)±\sqrt{\frac{4}{9}-4\left(-\frac{5}{4}\right)}}{2}$$

-4 ni -\frac{5}{4} marotabaga ko'paytirish.

$$u=\frac{-\left(-\frac{2}{3}\right)±\sqrt{\frac{4}{9}+5}}{2}$$

\frac{4}{9} ni 5 ga qo'shish.

$$u=\frac{-\left(-\frac{2}{3}\right)±\sqrt{\frac{49}{9}}}{2}$$

\frac{49}{9} ning kvadrat ildizini chiqarish.

$$u=\frac{-\left(-\frac{2}{3}\right)±\frac{7}{3}}{2}$$

-\frac{2}{3} ning teskarisi \frac{2}{3} ga teng.

$$u=\frac{\frac{2}{3}±\frac{7}{3}}{2}$$

u=\frac{\frac{2}{3}±\frac{7}{3}}{2} tenglamasini yeching, bunda ± musbat. Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{2}{3} ni \frac{7}{3} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$u=\frac{3}{2}$$

u=\frac{\frac{2}{3}±\frac{7}{3}}{2} tenglamasini yeching, bunda ± manfiy. Umumiy maxrajni topib va suratlarni ayirib \frac{7}{3} ni \frac{2}{3} dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$u=-\frac{\frac{5}{3}}{2}$$

-\frac{5}{3} ni 2 ga bo'lish.

$$u=-\frac{5}{6}$$

Tenglama yechildi.

$$u=\frac{3}{2} u=-\frac{5}{6}$$

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.

$$u^{2}-\frac{2}{3}u=\frac{5}{4}$$

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

$$u^{2}-\frac{2}{3}u+\left(-\frac{1}{3}\right)^{2}=\frac{5}{4}+\left(-\frac{1}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{3} kvadratini chiqarish.

$$u^{2}-\frac{2}{3}u+\frac{1}{9}=\frac{5}{4}+\frac{1}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{4} ni \frac{1}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$u^{2}-\frac{2}{3}u+\frac{1}{9}=\frac{49}{36}$$

u^{2}-\frac{2}{3}u+\frac{1}{9} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(u-\frac{1}{3}\right)^{2}=\frac{49}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(u-\frac{1}{3}\right)^{2}}=\sqrt{\frac{49}{36}}$$

Qisqartirish.

$$u-\frac{1}{3}=\frac{7}{6} u-\frac{1}{3}=-\frac{7}{6}$$

\frac{1}{3} ni tenglamaning ikkala tarafiga qo'shish.

$$u=\frac{3}{2} u=-\frac{5}{6}$$