1/15x^2-3/10x+1/3=0 eching

$$\frac { 1 } { 15 } x ^ { 2 } - \frac { 3 } { 10 } x + \frac { 1 } { 3 } = 0$$

x uchun yechish

$$x=2$$

$$x = \frac{5}{2} = 2\frac{1}{2} = 2,5$$

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.

$$\frac{1}{15}x^{2}-\frac{3}{10}x+\frac{1}{3}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} \frac{1}{15} ni a, -\frac{3}{10} ni b va \frac{1}{3} ni c bilan almashtiring.

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{\left(-\frac{3}{10}\right)^{2}-4\times \frac{1}{15}\times \frac{1}{3}}}{2\times \frac{1}{15}}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{10} kvadratini chiqarish.

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{\frac{9}{100}-4\times \frac{1}{15}\times \frac{1}{3}}}{2\times \frac{1}{15}}$$

-4 ni \frac{1}{15} marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{\frac{9}{100}-\frac{4}{15}\times \frac{1}{3}}}{2\times \frac{1}{15}}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali -\frac{4}{15} ni \frac{1}{3} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{\frac{9}{100}-\frac{4}{45}}}{2\times \frac{1}{15}}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{9}{100} ni -\frac{4}{45} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{\frac{1}{900}}}{2\times \frac{1}{15}}$$

\frac{1}{900} ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-\frac{3}{10}\right)±\frac{1}{30}}{2\times \frac{1}{15}}$$

-\frac{3}{10} ning teskarisi \frac{3}{10} ga teng.

$$x=\frac{\frac{3}{10}±\frac{1}{30}}{2\times \frac{1}{15}}$$

2 ni \frac{1}{15} marotabaga ko'paytirish.

$$x=\frac{\frac{3}{10}±\frac{1}{30}}{\frac{2}{15}}$$

x=\frac{\frac{3}{10}±\frac{1}{30}}{\frac{2}{15}} tenglamasini yeching, bunda ± musbat. Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{10} ni \frac{1}{30} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{\frac{1}{3}}{\frac{2}{15}}$$

\frac{1}{3} ni \frac{2}{15} ga bo'lish \frac{1}{3} ga k'paytirish \frac{2}{15} ga qaytarish.

$$x=\frac{5}{2}$$

x=\frac{\frac{3}{10}±\frac{1}{30}}{\frac{2}{15}} tenglamasini yeching, bunda ± manfiy. Umumiy maxrajni topib va suratlarni ayirib \frac{1}{30} ni \frac{3}{10} dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$x=\frac{\frac{4}{15}}{\frac{2}{15}}$$

\frac{4}{15} ni \frac{2}{15} ga bo'lish \frac{4}{15} ga k'paytirish \frac{2}{15} ga qaytarish.

$$x=2$$

Tenglama yechildi.

$$x=\frac{5}{2} x=2$$

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.

$$\frac{1}{15}x^{2}-\frac{3}{10}x+\frac{1}{3}=0$$

Tenglamaning ikkala tarafidan \frac{1}{3} ni ayirish.

$$\frac{1}{15}x^{2}-\frac{3}{10}x+\frac{1}{3}-\frac{1}{3}=-\frac{1}{3}$$

O‘zidan \frac{1}{3} ayirilsa 0 qoladi.

$$\frac{1}{15}x^{2}-\frac{3}{10}x=-\frac{1}{3}$$

Ikkala tarafini 15 ga ko‘paytiring.

$$\frac{\frac{1}{15}x^{2}-\frac{3}{10}x}{\frac{1}{15}}=-\frac{\frac{1}{3}}{\frac{1}{15}}$$

\frac{1}{15} ga bo'lish \frac{1}{15} ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{\frac{3}{10}}{\frac{1}{15}}\right)x=-\frac{\frac{1}{3}}{\frac{1}{15}}$$

-\frac{3}{10} ni \frac{1}{15} ga bo'lish -\frac{3}{10} ga k'paytirish \frac{1}{15} ga qaytarish.

$$x^{2}-\frac{9}{2}x=-\frac{\frac{1}{3}}{\frac{1}{15}}$$

-\frac{1}{3} ni \frac{1}{15} ga bo'lish -\frac{1}{3} ga k'paytirish \frac{1}{15} ga qaytarish.

$$x^{2}-\frac{9}{2}x=-5$$

-\frac{9}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{9}{4} olish uchun. Keyin, -\frac{9}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{9}{2}x+\left(-\frac{9}{4}\right)^{2}=-5+\left(-\frac{9}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{4} kvadratini chiqarish.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=-5+\frac{81}{16}$$

-5 ni \frac{81}{16} ga qo'shish.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{1}{16}$$

x^{2}-\frac{9}{2}x+\frac{81}{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.

$$\left(x-\frac{9}{4}\right)^{2}=\frac{1}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Qisqartirish.

$$x-\frac{9}{4}=\frac{1}{4} x-\frac{9}{4}=-\frac{1}{4}$$

\frac{9}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{5}{2} x=2$$