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## Algebra I

Bu yerda biz kvadrat uchhadni maxsus formulalar orqali birhad shakliga olib kelishni oʻrganamiz. Bu yigʻindi va ayirmaning kvadrati usulini koʻrsatadi, demak, siz ishga kirishishdan oldin buni bilishingiz kerak.

## Kirish: Qisqa koʻpaytirish formulalari bilan koʻpaytuvchilarga ajratish

Kvadratga koʻtarish uchun biz quyidagi formulalarni ishlatishimiz mumkin.
• left parenthesis, start color #11accd, a, end color #11accd, plus, start color #1fab54, b, end color #1fab54, right parenthesis, squared, equals, start color #11accd, a, end color #11accd, squared, plus, 2, start color #11accd, a, end color #11accd, start color #1fab54, b, end color #1fab54, plus, start color #1fab54, b, end color #1fab54, squared
• left parenthesis, start color #11accd, a, end color #11accd, minus, start color #1fab54, b, end color #1fab54, right parenthesis, squared, equals, start color #11accd, a, end color #11accd, squared, minus, 2, start color #11accd, a, end color #11accd, start color #1fab54, b, end color #1fab54, plus, start color #1fab54, b, end color #1fab54, squared
Eʼtibor bering, formuladagi a va b istalgan algebraik ifoda boʻlishi mumkin. Misol uchun, biz left parenthesis, x, plus, 5, right parenthesis, squared ni koʻpaytuvchilarga ajratmoqchimiz. Bunda start color #11accd, a, end color #11accd, equals, start color #11accd, x, end color #11accd va start color #1fab54, b, end color #1fab54, equals, start color #1fab54, 5, end color #1fab54 boʻlib, quyidagi natija hosil boʻladi:
\begin{aligned}(\blueD x+\greenD 5)^2&=\blueD x^2+2(\blueD x)(\greenD5)+(\greenD 5)^2\\\\ &=x^2+10x+25\end{aligned}
Siz bu formulani left parenthesis, x, plus, 5, right parenthesis, squared ni koʻpaytuvchilarga ajratib tekshirishingiz mumkin.
Bu kengaytmani ortiga qaytarish formulaga solishdir. Agar biz ifodani ortiga qaytarsak, koʻphadlarni koʻpaytuvchilarga ajratish formulasini hosil qilamiz: a, squared, plus minus, 2, a, b, plus, b, squared.
• start color #11accd, a, end color #11accd, squared, plus, 2, start color #11accd, a, end color #11accd, start color #1fab54, b, end color #1fab54, plus, start color #1fab54, b, end color #1fab54, squared, space, equals, left parenthesis, start color #11accd, a, end color #11accd, plus, start color #1fab54, b, end color #1fab54, right parenthesis, squared
• start color #11accd, a, end color #11accd, squared, minus, 2, start color #11accd, a, end color #11accd, start color #1fab54, b, end color #1fab54, plus, start color #1fab54, b, end color #1fab54, squared, space, equals, left parenthesis, start color #11accd, a, end color #11accd, minus, start color #1fab54, b, end color #1fab54, right parenthesis, squared
Biz birinchi formulani mana bu x, squared, plus, 10, x, plus, 25 koʻphad uchun ishlatsak boʻladi. Bizga start color #11accd, a, end color #11accd, equals, start color #11accd, x, end color #11accd va start color #1fab54, b, end color #1fab54, equals, start color #1fab54, 5, end color #1fab54 berilgan.
\begin{aligned}x^2+10x+25&=\blueD x^2+2(\blueD x)(\greenD5)+(\greenD 5)^2\\\\ &=(\blueD x+\greenD 5)^2\end{aligned}
Endi bir nechta qisqa koʻpaytirish formulalari qoʻllangan misollarni koʻrib chiqamiz.

## 1-misol: $x^2+8x+16$x, squared, plus, 8, x, plus, 16 ni koʻpaytuvchilarga ajratish

Eʼtibor bering, birinchi va soʻnggi hadlar kvadrat shaklda: x, squared, equals, left parenthesis, start color #11accd, x, end color #11accd, right parenthesis, squared va 16, equals, left parenthesis, start color #1fab54, 4, end color #1fab54, right parenthesis, squared. Oʻrtadagi had esa birinchi va soʻnggi hadlar koʻpaytmasining ikkilangan shaklida: 2, left parenthesis, start color #11accd, x, end color #11accd, right parenthesis, left parenthesis, start color #1fab54, 4, end color #1fab54, right parenthesis, equals, 8, x.
start color #11accd, a, end color #11accd, squared, plus, 2, start color #11accd, a, end color #11accd, start color #1fab54, b, end color #1fab54, plus, start color #1fab54, b, end color #1fab54, squared, space, equals, left parenthesis, start color #11accd, a, end color #11accd, plus, start color #1fab54, b, end color #1fab54, right parenthesis, squared
start color #11accd, a, end color #11accd, equals, start color #11accd, x, end color #11accd va start color #1fab54, b, end color #1fab54, equals, start color #1fab54, 4, end color #1fab54 berilgan. Biz bu koʻphadni quyidagi tarzda koʻpaytuvchilarga ajratishimiz mumkin:
\begin{aligned}x^2+8x+16&=(\blueD x)^2+2(\blueD x)(\greenD 4)+(\greenD4)^2\\ \\ &=(\blueD{x}+\greenD{4})^2\end{aligned}
Biz misolni left parenthesis, x, plus, 4, right parenthesis, squared ni ochib chiqib tekshirishimiz mumkin:
\begin{aligned}(x+4)^2&=(x)^2+2(x)(4)+(4)^2\\ \\ &=x^2+8x+16 \end{aligned}

### Mavzu boʻyicha bilimingizni tekshiring

1) x, squared, plus, 6, x, plus, 9 ni koʻpaytuvchilarga ajrating.
Bitta javobni tanlang:

2) x, squared, minus, 6, x, plus, 9 ni koʻpaytuvchilarga ajrating.
Bitta javobni tanlang:

3) x, squared, plus, 14, x, plus, 49 ni koʻpaytuvchilarga ajrating.

## 2-misol: $4x^2+12x+9$4, x, squared, plus, 12, x, plus, 9 ni koʻpaytuvchilarga ajratish

Qisqa koʻpaytirish formulalarida birinchi hadning koeffitsiyenti 1 ga teng boʻlishi shart emas.
Misol uchun, 4, x, squared, plus, 12, x, plus, 9 da birinchi va soʻnggi hadlar kvadrat shaklda: 4, x, squared, equals, left parenthesis, start color #11accd, 2, x, end color #11accd, right parenthesis, squared va 9, equals, left parenthesis, start color #1fab54, 3, end color #1fab54, right parenthesis, squared. Oʻrtadagi had esa birinchi va soʻnggi hadlar koʻpaytmasining ikkilangan shaklida: 2, left parenthesis, start color #11accd, 2, x, end color #11accd, right parenthesis, left parenthesis, start color #1fab54, 3, end color #1fab54, right parenthesis, equals, 12, x.
Chunki, 4, x, squared, plus, 12, x, plus, 9 ifoda ham qisqa koʻpaytirish formulalariga tushadi. Biz yana quyidagi formulani ishlatishimiz mumkin.
start color #11accd, a, end color #11accd, squared, plus, 2, start color #11accd, a, end color #11accd, start color #1fab54, b, end color #1fab54, plus, start color #1fab54, b, end color #1fab54, squared, space, equals, left parenthesis, start color #11accd, a, end color #11accd, plus, start color #1fab54, b, end color #1fab54, right parenthesis, squared
Bu holda, start color #11accd, a, end color #11accd, equals, start color #11accd, 2, x, end color #11accd va start color #1fab54, b, end color #1fab54, equals, start color #1fab54, 3, end color #1fab54. Koʻphad quyidagi shaklga keladi:
\begin{aligned}4x^2+12x+9&=(\blueD {2x})^2+2(\blueD {2x})(\greenD 3)+(\greenD3)^2\\ \\ &=(\blueD{2x}+\greenD{3})^2\end{aligned}
Biz misolni left parenthesis, 2, x, plus, 3, right parenthesis, squared ifodani koʻpaytuvchilarga ajratib tekshirishimiz mumkin.

### Tushunganingizni tekshiring.

4) 9, x, squared, plus, 30, x, plus, 25 ni koʻpaytuvchilarga ajrating.
Bitta javobni tanlang:

5) 4, x, squared, minus, 20, x, plus, 25 ni koʻpaytuvchilarga ajrating.

## Murakkab masalalar

6*) x, start superscript, 4, end superscript, plus, 2, x, squared, plus, 1 ni koʻpaytuvchilarga ajrating.

7*) 9, x, squared, plus, 24, x, y, plus, 16, y, squared ni koʻpaytuvchilarga ajrating.