25x^2-90x+77=0 eching

$$25 { x }^{ 2 } -90x+77=0$$

x uchun yechish

$$x = \frac{7}{5} = 1\frac{2}{5} = 1,4$$

$$x = \frac{11}{5} = 2\frac{1}{5} = 2,2$$

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.

$$25x^{2}-90x+77=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 25 ni a, -90 ni b va 77 ni c bilan almashtiring.

$$x=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\times 25\times 77}}{2\times 25}$$

-90 kvadratini chiqarish.

$$x=\frac{-\left(-90\right)±\sqrt{8100-4\times 25\times 77}}{2\times 25}$$

-4 ni 25 marotabaga ko'paytirish.

$$x=\frac{-\left(-90\right)±\sqrt{8100-100\times 77}}{2\times 25}$$

-100 ni 77 marotabaga ko'paytirish.

$$x=\frac{-\left(-90\right)±\sqrt{8100-7700}}{2\times 25}$$

8100 ni -7700 ga qo'shish.

$$x=\frac{-\left(-90\right)±\sqrt{400}}{2\times 25}$$

400 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-90\right)±20}{2\times 25}$$

-90 ning teskarisi 90 ga teng.

$$x=\frac{90±20}{2\times 25}$$

2 ni 25 marotabaga ko'paytirish.

$$x=\frac{90±20}{50}$$

x=\frac{90±20}{50} tenglamasini yeching, bunda ± musbat. 90 ni 20 ga qo'shish.

$$x=\frac{110}{50}$$

\frac{110}{50} ulushini 10 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{11}{5}$$

x=\frac{90±20}{50} tenglamasini yeching, bunda ± manfiy. 90 dan 20 ni ayirish.

$$x=\frac{70}{50}$$

\frac{70}{50} ulushini 10 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{7}{5}$$

Tenglama yechildi.

$$x=\frac{11}{5} x=\frac{7}{5}$$

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.

$$25x^{2}-90x+77=0$$

Tenglamaning ikkala tarafidan 77 ni ayirish.

$$25x^{2}-90x+77-77=-77$$

O‘zidan 77 ayirilsa 0 qoladi.

$$25x^{2}-90x=-77$$

Ikki tarafini 25 ga bo‘ling.

$$\frac{25x^{2}-90x}{25}=-\frac{77}{25}$$

25 ga bo'lish 25 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{90}{25}\right)x=-\frac{77}{25}$$

\frac{-90}{25} ulushini 5 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{18}{5}x=-\frac{77}{25}$$

-\frac{18}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{9}{5} olish uchun. Keyin, -\frac{9}{5} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{18}{5}x+\left(-\frac{9}{5}\right)^{2}=-\frac{77}{25}+\left(-\frac{9}{5}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{5} kvadratini chiqarish.

$$x^{2}-\frac{18}{5}x+\frac{81}{25}=\frac{-77+81}{25}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{77}{25} ni \frac{81}{25} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{18}{5}x+\frac{81}{25}=\frac{4}{25}$$

x^{2}-\frac{18}{5}x+\frac{81}{25} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{9}{5}\right)^{2}=\frac{4}{25}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{9}{5}\right)^{2}}=\sqrt{\frac{4}{25}}$$

Qisqartirish.

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

\frac{9}{5} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{11}{5} x=\frac{7}{5}$$