2019x^2=2020+[x] eching

$$2019 x ^ { 2 } = 2020 + [ x ]$$

x uchun yechish

$$x=-1$$

$$x = \frac{2020}{2019} = 1\frac{1}{2019} \approx 1,000495295$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan 2020 ni ayirish.

$$2019x^{2}-2020=x$$

Ikkala tarafdan x ni ayirish.

$$2019x^{2}-2020-x=0$$

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$2019x^{2}-x-2020=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 2019x^{2}+ax+bx-2020 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-1 ab=2019\left(-2020\right)=-4078380$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -4078380-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-4078380 2,-2039190 3,-1359460 4,-1019595 5,-815676 6,-679730 10,-407838 12,-339865 15,-271892 20,-203919 30,-135946 60,-67973 101,-40380 202,-20190 303,-13460 404,-10095 505,-8076 606,-6730 673,-6060 1010,-4038 1212,-3365 1346,-3030 1515,-2692 2019,-2020$$

Har bir juftlik yigʻindisini hisoblang.

$$1-4078380=-4078379 2-2039190=-2039188 3-1359460=-1359457 4-1019595=-1019591 5-815676=-815671 6-679730=-679724 10-407838=-407828 12-339865=-339853 15-271892=-271877 20-203919=-203899 30-135946=-135916 60-67973=-67913 101-40380=-40279 202-20190=-19988 303-13460=-13157 404-10095=-9691 505-8076=-7571 606-6730=-6124 673-6060=-5387 1010-4038=-3028 1212-3365=-2153 1346-3030=-1684 1515-2692=-1177 2019-2020=-1$$

Yechim – -1 yigʻindisini beruvchi juftlik.

$$a=-2020 b=2019$$

2019x^{2}-x-2020 ni \left(2019x^{2}-2020x\right)+\left(2019x-2020\right) sifatida qaytadan yozish.

$$\left(2019x^{2}-2020x\right)+\left(2019x-2020\right)$$

2019x^{2}-2020x ichida x ni ajrating.

$$x\left(2019x-2020\right)+2019x-2020$$

Distributiv funktsiyasidan foydalangan holda 2019x-2020 umumiy terminini chiqaring.

$$\left(2019x-2020\right)\left(x+1\right)$$

Tenglamani yechish uchun 2019x-2020=0 va x+1=0 ni yeching.

$$x=\frac{2020}{2019} x=-1$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 2020 ni ayirish.

$$2019x^{2}-2020=x$$

Ikkala tarafdan x ni ayirish.

$$2019x^{2}-2020-x=0$$

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.

$$2019x^{2}-x-2020=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2019 ni a, -1 ni b va -2020 ni c bilan almashtiring.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 2019\left(-2020\right)}}{2\times 2019}$$

-4 ni 2019 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1-8076\left(-2020\right)}}{2\times 2019}$$

-8076 ni -2020 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1+16313520}}{2\times 2019}$$

1 ni 16313520 ga qo'shish.

$$x=\frac{-\left(-1\right)±\sqrt{16313521}}{2\times 2019}$$

16313521 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-1\right)±4039}{2\times 2019}$$

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±4039}{2\times 2019}$$

2 ni 2019 marotabaga ko'paytirish.

$$x=\frac{1±4039}{4038}$$

x=\frac{1±4039}{4038} tenglamasini yeching, bunda ± musbat. 1 ni 4039 ga qo'shish.

$$x=\frac{4040}{4038}$$

\frac{4040}{4038} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{2020}{2019}$$

x=\frac{1±4039}{4038} tenglamasini yeching, bunda ± manfiy. 1 dan 4039 ni ayirish.

$$x=-\frac{4038}{4038}$$

-4038 ni 4038 ga bo'lish.

$$x=-1$$

Tenglama yechildi.

$$x=\frac{2020}{2019} x=-1$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan x ni ayirish.

$$2019x^{2}-x=2020$$

Ikki tarafini 2019 ga bo‘ling.

$$\frac{2019x^{2}-x}{2019}=\frac{2020}{2019}$$

2019 ga bo'lish 2019 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{1}{2019}x=\frac{2020}{2019}$$

-\frac{1}{2019} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{4038} olish uchun. Keyin, -\frac{1}{4038} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1}{2019}x+\left(-\frac{1}{4038}\right)^{2}=\frac{2020}{2019}+\left(-\frac{1}{4038}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{4038} kvadratini chiqarish.

$$x^{2}-\frac{1}{2019}x+\frac{1}{16305444}=\frac{2020}{2019}+\frac{1}{16305444}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{2020}{2019} ni \frac{1}{16305444} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{2019}x+\frac{1}{16305444}=\frac{16313521}{16305444}$$

x^{2}-\frac{1}{2019}x+\frac{1}{16305444} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{4038}\right)^{2}=\frac{16313521}{16305444}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{4038}\right)^{2}}=\sqrt{\frac{16313521}{16305444}}$$

Qisqartirish.

$$x-\frac{1}{4038}=\frac{4039}{4038} x-\frac{1}{4038}=-\frac{4039}{4038}$$

\frac{1}{4038} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{2020}{2019} x=-1$$