EURASIAN JOURNAL OF MATHEMATICAL
THEORY AND COMPUTER SCIENCES
Innovative Academy Research Support Center
Volume 5 Issue 6, June 2025 ISSN 2181-2861
Page 13
EFFECTIVE METHODS FOR SOLVING QUADRATIC
EQUATIONS USING GEOMETRIC METHODS
Abduraxmonov Baxromjon Alisherovich
1
Ochilova Aziza Yorqul qizi
2
1.Head of the Department of “Physics, Mathematics and Information
Technologies” of the Tashkent Pharmaceutical Institute, Doctor of
Physical and Mathematical Sciences, Associate Professor
2. Assistant of the Department of "Physics, Mathematics and
Information Technologies" of the Tashkent Pharmaceutical Institute
https://doi.org/10.5281/zenodo.15690306
ARTICLE INFO
ABSTRACT
Received: 12
th
June 2025
Accepted: 17
th
June 2025
Online: 18
th
June 2025
This article analyzes the possibilities of using geometric
methods in solving quadratic equations. These approaches are
important not only for strengthening theoretical knowledge,
but also for solving practical problems.
KEYWORDS
Quadratic
equation,
geometric method, straight
line, solution, surface.
ЭФФЕКТИВНЫЕ МЕТОДЫ РЕШЕНИЯ КВАДРАТНЫХ УРАВНЕНИЙ С
ИСПОЛЬЗОВАНИЕМ ГЕОМЕТРИЧЕСКИХ МЕТОДОВ
Абдурахмонов Бахромжон Алишерович
1
Очилова Азиза Ёркул кизи
2
1.Заведующий кафедрой «Физика, математика и информационные технологии»
Ташкентского фармацевтического института, кандидат физ.-мат. наук, доцент
2.Ассистент кафедры физики, математики и информационных технологий
Ташкентского фармацевтического института
https://doi.org/10.5281/zenodo.15690306
ARTICLE INFO
ABSTRACT
Received: 12
th
June 2025
Accepted: 17
th
June 2025
Online: 18
th
June 2025
В данной статье анализируются возможности
использования геометрических методов при решении
квадратных уравнений. Эти подходы важны не только
для закрепления теоретических знаний, но и для решения
практических задач.
KEYWORDS
Квадратное уравнение,
геометрический метод,
прямая,
решение,
поверхность.
GEOMETRIK USULLAR YORDAMIDA KVADRAT TENGLAMALARNI
YECHISHNING SAMARALI USULLARI
Abduraxmonov Baxromjon Alisherovich
1
Ochilova Aziza Yorqul qizi
2
1.Toshkent farmatsevtika instituti “Fizika, matematika va axborot texnologiyalari”
kafedrasi mudiri, f.-m.f.n., dotsent
2.Toshkent farmatsevtika instituti “Fizika, matematika va axborot texnologiyalari”
kafedrasi assistenti
https://doi.org/10.5281/zenodo.15690306
EURASIAN JOURNAL OF MATHEMATICAL
THEORY AND COMPUTER SCIENCES
Innovative Academy Research Support Center
Volume 5 Issue 6, June 2025 ISSN 2181-2861
Page 14
ARTICLE INFO
ABSTRACT
Received: 12
th
June 2025
Accepted: 17
th
June 2025
Online: 18
th
June 2025
Ushbu maqolada kvadrat tenglamalarni yechishda geometrik
usullardan foydalanish imkoniyatlari tahlil qilinadi. Ushbu
yondashuvlar nafaqat nazariy bilimlarni mustahkamlashga,
balki amaliy masalalarni yechishda ham muhim ahamiyat
kasb etadi.
KEYWORDS
Kvadrat
tenglama,
geometrik
usul,
to‘g‘ri
chiziq, yechim, yuza.
Matematikada kvadrat tenglamalarni yechishning turli xil usullari mavjud bo‘lib, ulardan
algebraik va geometrik yondashuvlar keng qo‘llaniladi. Algebraik usullar odatda formulalar va
hisob-kitoblarga asoslangan bo‘lsa, geometrik usullar esa tushunchalarni vizual tarzda
ifodalash va muammolarni grafik yoki shakllar yordamida yechishga imkon beradi.
Geometrik yondashuv kvadrat tenglamalarning ildizlarini tushunishni osonlashtiradi va
ularni grafik shaklda tasvirlash orqali yechish imkoniyatini yaratadi. Ayniqsa, parabola, to‘g‘ri
chiziq va boshqa geometrik shakllar yordamida kvadrat tenglamalarni yechish usuli matematik
tushunchalarni yanada chuqurroq anglashga yordam beradi.
1-misol.
Kvadrat tenglamaning yechimini toping.
2
10
39
x
x
Yechish:
Bu misol quyidagicha ifodalangan. Yon tomonlari
x
bo‘lgan kichik kvadrat
chizamiz (1-rasmdagi qora rangda) va uning yon tomonlariga balandligi
10
4
ga teng bo‘lgan
to‘g‘ri to‘rtburchak (dastlabki tenglamada
x
uchun koeffitsyent 10 ga teng) shu shaklning
burchaklariga tomonlari
10
4
bo‘lgan kvadrat chizamiz.
1-
rasm
Katta kvadratning yuzasi barcha to‘g‘ri to‘rtburchaklar yuzalari yig‘indisiga teng.
2
2
2
2
10
10
10
4
10
4
4
4
4
x
x
x
x
Shart bo‘yicha
2
10
39
x
x
, ya’ni katta kvadratning maydoni
2
10
39
4 39 25 64
4
EURASIAN JOURNAL OF MATHEMATICAL
THEORY AND COMPUTER SCIENCES
Innovative Academy Research Support Center
Volume 5 Issue 6, June 2025 ISSN 2181-2861
Page 15
Demak, katta kvadratning tomoni 8 ga teng. Shuning uchun
10
2
8
4
x
bo’lsa,
x
ning qiymati
3
x
bo‘ladi.
2-misol.
Kvadrat tenglamani yeching.
2
6
16 0
y
y
.
Yechish.
Kvadrat chizamiz. (2-rasm)
2-
rasm
Dastlabki tenglamaning shartlarini qayta joylashtiramiz:
2
6
16
y
y
Tenglamalar
yechimining xossalariga asoslanib
2
6
9 16 9
y
y
yoki
2
6
9 25
y
y
. Bu shuni
anglatadiki,
2
6
9
y
y
va
25
lar geometrik jihatdan yon tomonlari
3
y
bo‘lgan
kvadratning yuzasi, ya’ni yig‘indisi ushbu maydonni tashkil etuvchi to‘g‘ri to‘rtburchak
maydonlaridan bir tomondan
2
2 3
9
y
y
va bosha tamondan
2
3
y
Shunday qilib,
ushbu geometrik nisbat algebraik ravishda quyidagicha qayta yozilishi mumkin:
2
2
6
9
3
y
y
y
,
3
5
y
1
2
y
,
2
8
y
3-misol.
Kvadrat tenglamani yeching.
2
10
16
0
x
x
Yechish:
Tomoni
x
bo‘lgan bo‘lgan kvadrat va uning ichida tomonlari
x
va 5 bo‘lgan
ikkita to‘g‘ri to‘rtburchak chizamiz. (3-rasm)
Tomoni
5
x
bo‘lgan ichki kvadratning maydoni
2
5
5
25
S
x
x
x
ga teng
bo‘ladi. Dastlabki tenglamadan
2
10
16
x
x
. U holda
25 16 9
S
va ichki kvadratning
tomoni 3 ga teng.
3-
rasm
EURASIAN JOURNAL OF MATHEMATICAL
THEORY AND COMPUTER SCIENCES
Innovative Academy Research Support Center
Volume 5 Issue 6, June 2025 ISSN 2181-2861
Page 16
Ushbu usuldan foydalanib siz
2
x
px
q
ko‘rinishidagi kvadrat tenglamaning musbat
ildizlarini topishingiz mumkin.
Kvadrat tenglamalarni yechishda geometrik usullarning qo‘llanilishi matematikaning
algebraik va geometrik yo‘nalishlari o‘rtasidagi o‘zaro bog‘liqlikni yanada chuqurroq anglash
imkonini beradi. Shu bois, kvadrat tenglamalarni yechishda geometrik yondashuvning
qo‘llanilishi nafaqat nazariy jihatdan muhim ahamiyat kasb etadi, balki u amaliy hisob-kitoblar,
muhandislik, fizika va boshqa tabiiy fanlar sohasida ham samarali qo‘llanilishi mumkin. Ushbu
usullar orqali algebra va geometriya o‘rtasidagi bog‘liqlik aniqroq ko‘rinadi. Shuning uchun
maktab va litsey o‘quvchilari kvadrat tenglamalarni faqat formulalar orqali emas, balki
geometrik usullar bilan ham yechishga odatlanishlari foydalidir. Bu nafaqat darslarni yaxshiroq
tushunishga, balki matematikaga bo‘lgan qiziqishni oshirishga ham yordam beradi.
References:
1.
Организационно-методическое обеспечение учебного процесса в вузе. Учебно-
методическое пособие / Н. Г. Берденникова, В. И. Меденцев. – СПб: БАТиП, 2006.
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https://www.doi.org/10.5281/zenodo.7813169
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