50/49x^2-10/49x-24/49=0 eching

$$\frac{ 50 }{ 49 } { x }^{ 2 } - \frac{ 10 }{ 49 } x- \frac{ 24 }{ 49 } = 0$$

x uchun yechish

$$x=-\frac{3}{5}=-0,6$$

$$x=\frac{4}{5}=0,8$$

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.

$$\frac{50}{49}x^{2}-\frac{10}{49}x-\frac{24}{49}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} \frac{50}{49} ni a, -\frac{10}{49} ni b va -\frac{24}{49} ni c bilan almashtiring.

$$x=\frac{-\left(-\frac{10}{49}\right)±\sqrt{\left(-\frac{10}{49}\right)^{2}-4\times \frac{50}{49}\left(-\frac{24}{49}\right)}}{2\times \frac{50}{49}}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{10}{49} kvadratini chiqarish.

$$x=\frac{-\left(-\frac{10}{49}\right)±\sqrt{\frac{100}{2401}-4\times \frac{50}{49}\left(-\frac{24}{49}\right)}}{2\times \frac{50}{49}}$$

-4 ni \frac{50}{49} marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{10}{49}\right)±\sqrt{\frac{100}{2401}-\frac{200}{49}\left(-\frac{24}{49}\right)}}{2\times \frac{50}{49}}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali -\frac{200}{49} ni -\frac{24}{49} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$x=\frac{-\left(-\frac{10}{49}\right)±\sqrt{\frac{100+4800}{2401}}}{2\times \frac{50}{49}}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{100}{2401} ni \frac{4800}{2401} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{-\left(-\frac{10}{49}\right)±\sqrt{\frac{100}{49}}}{2\times \frac{50}{49}}$$

\frac{100}{49} ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-\frac{10}{49}\right)±\frac{10}{7}}{2\times \frac{50}{49}}$$

-\frac{10}{49} ning teskarisi \frac{10}{49} ga teng.

$$x=\frac{\frac{10}{49}±\frac{10}{7}}{2\times \frac{50}{49}}$$

2 ni \frac{50}{49} marotabaga ko'paytirish.

$$x=\frac{\frac{10}{49}±\frac{10}{7}}{\frac{100}{49}}$$

x=\frac{\frac{10}{49}±\frac{10}{7}}{\frac{100}{49}} tenglamasini yeching, bunda ± musbat. Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{10}{49} ni \frac{10}{7} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{\frac{80}{49}}{\frac{100}{49}}$$

\frac{80}{49} ni \frac{100}{49} ga bo'lish \frac{80}{49} ga k'paytirish \frac{100}{49} ga qaytarish.

$$x=\frac{4}{5}$$

x=\frac{\frac{10}{49}±\frac{10}{7}}{\frac{100}{49}} tenglamasini yeching, bunda ± manfiy. Umumiy maxrajni topib va suratlarni ayirib \frac{10}{7} ni \frac{10}{49} dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$x=-\frac{\frac{60}{49}}{\frac{100}{49}}$$

-\frac{60}{49} ni \frac{100}{49} ga bo'lish -\frac{60}{49} ga k'paytirish \frac{100}{49} ga qaytarish.

$$x=-\frac{3}{5}$$

Tenglama yechildi.

$$x=\frac{4}{5} x=-\frac{3}{5}$$

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.

$$\frac{50}{49}x^{2}-\frac{10}{49}x-\frac{24}{49}=0$$

\frac{24}{49} ni tenglamaning ikkala tarafiga qo'shish.

$$\frac{50}{49}x^{2}-\frac{10}{49}x-\frac{24}{49}-\left(-\frac{24}{49}\right)=-\left(-\frac{24}{49}\right)$$

O‘zidan -\frac{24}{49} ayirilsa 0 qoladi.

$$\frac{50}{49}x^{2}-\frac{10}{49}x=-\left(-\frac{24}{49}\right)$$

0 dan -\frac{24}{49} ni ayirish.

$$\frac{50}{49}x^{2}-\frac{10}{49}x=\frac{24}{49}$$

Tenglamaning ikki tarafini \frac{50}{49} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$\frac{\frac{50}{49}x^{2}-\frac{10}{49}x}{\frac{50}{49}}=\frac{\frac{24}{49}}{\frac{50}{49}}$$

\frac{50}{49} ga bo'lish \frac{50}{49} ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{\frac{10}{49}}{\frac{50}{49}}\right)x=\frac{\frac{24}{49}}{\frac{50}{49}}$$

-\frac{10}{49} ni \frac{50}{49} ga bo'lish -\frac{10}{49} ga k'paytirish \frac{50}{49} ga qaytarish.

$$x^{2}-\frac{1}{5}x=\frac{\frac{24}{49}}{\frac{50}{49}}$$

\frac{24}{49} ni \frac{50}{49} ga bo'lish \frac{24}{49} ga k'paytirish \frac{50}{49} ga qaytarish.

$$x^{2}-\frac{1}{5}x=\frac{12}{25}$$

-\frac{1}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{10} olish uchun. Keyin, -\frac{1}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1}{5}x+\left(-\frac{1}{10}\right)^{2}=\frac{12}{25}+\left(-\frac{1}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{10} kvadratini chiqarish.

$$x^{2}-\frac{1}{5}x+\frac{1}{100}=\frac{12}{25}+\frac{1}{100}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{12}{25} ni \frac{1}{100} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{5}x+\frac{1}{100}=\frac{49}{100}$$

x^{2}-\frac{1}{5}x+\frac{1}{100} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{10}\right)^{2}=\frac{49}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{10}\right)^{2}}=\sqrt{\frac{49}{100}}$$

Qisqartirish.

$$x-\frac{1}{10}=\frac{7}{10} x-\frac{1}{10}=-\frac{7}{10}$$

\frac{1}{10} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{4}{5} x=-\frac{3}{5}$$