9x^2-245x+500=0 eching

$$9 { x }^{ 2 } -245x+500=0$$

x uchun yechish

$$x = \frac{20}{9} = 2\frac{2}{9} \approx 2,222222222$$

$$x=25$$

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.

$$9x^{2}-245x+500=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 9 ni a, -245 ni b va 500 ni c bilan almashtiring.

$$x=\frac{-\left(-245\right)±\sqrt{\left(-245\right)^{2}-4\times 9\times 500}}{2\times 9}$$

-245 kvadratini chiqarish.

$$x=\frac{-\left(-245\right)±\sqrt{60025-4\times 9\times 500}}{2\times 9}$$

-4 ni 9 marotabaga ko'paytirish.

$$x=\frac{-\left(-245\right)±\sqrt{60025-36\times 500}}{2\times 9}$$

-36 ni 500 marotabaga ko'paytirish.

$$x=\frac{-\left(-245\right)±\sqrt{60025-18000}}{2\times 9}$$

60025 ni -18000 ga qo'shish.

$$x=\frac{-\left(-245\right)±\sqrt{42025}}{2\times 9}$$

42025 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-245\right)±205}{2\times 9}$$

-245 ning teskarisi 245 ga teng.

$$x=\frac{245±205}{2\times 9}$$

2 ni 9 marotabaga ko'paytirish.

$$x=\frac{245±205}{18}$$

x=\frac{245±205}{18} tenglamasini yeching, bunda ± musbat. 245 ni 205 ga qo'shish.

$$x=\frac{450}{18}$$

450 ni 18 ga bo'lish.

$$x=25$$

x=\frac{245±205}{18} tenglamasini yeching, bunda ± manfiy. 245 dan 205 ni ayirish.

$$x=\frac{40}{18}$$

\frac{40}{18} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{20}{9}$$

Tenglama yechildi.

$$x=25 x=\frac{20}{9}$$

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.

$$9x^{2}-245x+500=0$$

Tenglamaning ikkala tarafidan 500 ni ayirish.

$$9x^{2}-245x+500-500=-500$$

O‘zidan 500 ayirilsa 0 qoladi.

$$9x^{2}-245x=-500$$

Ikki tarafini 9 ga bo‘ling.

$$\frac{9x^{2}-245x}{9}=-\frac{500}{9}$$

9 ga bo'lish 9 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{245}{9}x=-\frac{500}{9}$$

-\frac{245}{9} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{245}{18} olish uchun. Keyin, -\frac{245}{18} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{245}{9}x+\left(-\frac{245}{18}\right)^{2}=-\frac{500}{9}+\left(-\frac{245}{18}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{245}{18} kvadratini chiqarish.

$$x^{2}-\frac{245}{9}x+\frac{60025}{324}=-\frac{500}{9}+\frac{60025}{324}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{500}{9} ni \frac{60025}{324} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{245}{9}x+\frac{60025}{324}=\frac{42025}{324}$$

x^{2}-\frac{245}{9}x+\frac{60025}{324} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{245}{18}\right)^{2}=\frac{42025}{324}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{245}{18}\right)^{2}}=\sqrt{\frac{42025}{324}}$$

Qisqartirish.

$$x-\frac{245}{18}=\frac{205}{18} x-\frac{245}{18}=-\frac{205}{18}$$

\frac{245}{18} ni tenglamaning ikkala tarafiga qo'shish.

$$x=25 x=\frac{20}{9}$$