✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- (a + b)² = ?
- 50² = ?
- 7² − 3² = ?
Answers dekho
- a² + 2ab + b²
- 2500
- 40 (= 10 × 4!)
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 4 — (a + b)² Ka Formula dekh lijiye.
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🎬 Kahani: Darzi Ka Kapda Aur Do Pattiyaan
Darzi Iqbal chacha ke paas 50 cm ka square kapda tha. Unhe 48 cm ka square chahiye tha, to ek taraf aur neeche se 2 cm ki patti kaati. 🧵
Shagird bola: “Do pattiyaan 50 × 2 = 100-100, to 2500 − 200 = 2300 bacha.” Chacha muskuraye: “Kone ka tukda do baar kaat diya tune!”
Sahi jawab 2304. Woh 4 kahan se aaya? Aaj (a − b)² aur ek aur jaadu (a + b)(a − b) chitron se samjhenge. ✂️
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📖 Concept: Ghataav Mein Kone Ka Tukda Wapas Jodo
💡 Is post ki ek badi baat: a² mein se do pattiyaan (a × b) kaato to kone ka b × b tukda do baar kat jaata, isliye ek baar wapas jodo: (a − b)² = a² − 2ab + b². Aur jod × farak mein pattiyaan kat jaati: (a + b)(a − b) = a² − b².
Step 1: (a − b)²: chitra
a² − ab − ab kiya to kona (b²) do baar gaya. + b² karke sahi: 50² − 2 × 100 + 4 = 2304.
Check: 48 × 48 = 2304 ✅
Step 2: (a − b)²: algebra, do tareeke
A: (a − b)(a − b) = a² − ab − ba + b² = a² − 2ab + b².
B: (a + (−b))² = a² + 2a(−b) + (−b)² = a² − 2ab + b². (Post 4 — (a + b)² Ka Formula ki identity hi, b negative!)
Kab kaunsa? B dikhata ki naya kuch yaad karne ki zaroorat nahi. A tab jab sign ka confusion ho.
Step 3: Mann mein squares (ghataav se)
89² = (90 − 1)² = 8100 − 180 + 1 = 7921.
996² = (1000 − 4)² = 1000000 − 8000 + 16 = 992016.
Kab (+) kab (−)? Gol number ke upar ho (71) to (a + b)²; neeche (89) to (a − b)². Jo gol number paas ho, wahi chuno.
Step 4: (a + b)(a − b) = a² − b²
Distributive: a² − ab + ba − b² = a² − b² (beech wale kat gaye!).
67 × 73 = (70 − 3)(70 + 3) = 4900 − 9 = 4891. 104 × 96 = 10000 − 16 = 9984.
Chitra: a² mein se kone ka b² kaato. Bache L-shape ki ek patti kaat ke ghuma ke jodo: (a + b) × (a − b) ka rectangle!
Step 5: Sridharacharya ka square tareeka
a² = (a + b)(a − b) + b². b aisa chuno ki ek bracket gol number bane.
47² = 50 × 44 + 3² = 2200 + 9 = 2209. 193² = 200 × 186 + 49 = 37249.
WHY sahi? (a + b)(a − b) = a² − b², to b² wapas jodne se a².
🪜 Atak Gaye? Aise Socho
- Square poochha? Paas ka gol number dhundho.
- Upar ho to (a + b)², neeche ho to (a − b)².
- Do numbers ka guna? Kya dono ek number se barabar door? To a² − b².
- Signs: (−b)² = +b² hamesha.
- Jawab ko last digit se check (89² ka last digit 1).
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🔐 Deep Concept: Teen Identity, Ek Hi Parivaar
(a + b)², (a − b)², (a + b)(a − b): teeno (a + m)(b + n) (Post 2 — Guna Mein Badlaav: Increments) ke khaas case hain. Aur woh distributive (Post 1 — Distributive Property Kya Hai?) ka. Ek ped, teen daaliyaan! 🌳
Pattern 1: (a + b)² + (a − b)² = 2(a² + b²) (2ab aur −2ab kat jaate). 2(4² + 7²) = 130 = 11² + 3² ✅
Pattern 2: a² − b² = (a + b)(a − b): 12² − 5² = 119 = 17 × 7 ✅
(a − b)² aur (b − a)²: barabar! Kyunki (b − a) = −(a − b), aur minus ka square plus.
Unifying idea: identities alag-alag rataane ki cheez nahi: ek distributive, aur signs ka khel. 🎯
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✏️ Solved Examples
Example 1 (Easy): (2x − 5)² kholo.
4x² − 20x + 25.
Check: x = 3: 1² = 1; 36 − 60 + 25 = 1 ✅
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Example 2 (Medium): 38 × 42 mann mein.
(40 − 2)(40 + 2) = 1600 − 4 = 1596.
Check: 38 × 42 = 1520 + 76 = 1596 ✅
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Example 3 (NCERT style): 64 ko do squares ke farak mein likho (do tareeke).
a² − b² = (a + b)(a − b) = 64: 16 × 4 → a = 10, b = 6; 32 × 2 → a = 17, b = 15.
Check: 100 − 36 = 64 ✅, 289 − 225 = 64 ✅
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Iqbal chacha ka sabak: 2500 − 200 + 4 = 2304. Shagird ne kone ka tukda wapas joda. Chacha: “Kapda ho ya algebra, do baar kaata hua ek baar wapas lo!” ✂️
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🏠 (a − b)² Aur a² − b² Roz Ki Zindagi Mein
- 🧵 Darzi: kapde se border kaatna: (a − b)².
- 🏞️ Park ka raasta: bade square park mein andar ka square ghaas: raasta = a² − b².
- 🛒 Bazaar: ₹48 × 52 cheezein = 2500 − 4 = ₹2496.
- 🖼️ Frame: 30 cm frame, andar 26 cm photo: frame ka area 30² − 26² = 56 × 4 = 224.
- 🧮 Competitive exams: 997², 1003 × 997 jaise sawaal second mein.
Frame wala sawaal dekho: 900 − 676 lamba hisaab, par 56 × 4 turant! Yahi a² − b² ki taakat.
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🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Square, gol se neeche | ✅ (a − b)² | 89 = 90 − 1 |
| Square, gol se upar | ✅ (a + b)² | 71 = 70 + 1 |
| Do number, barabar door | ✅ a² − b² | 67 × 73 |
| Square koi bhi | ✅ Sridhara: (a + b)(a − b) + b² | Gol bracket banao |
| Farak of squares | ✅ (a + b)(a − b) | Ring/frame area |
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❌→✅ Worked Mistake
Sawaal: (7 − y)² kholo.
❌ Galat: 49 − y². Ya 49 − 14y − y².
✅ Sahi: 49 − 14y + y². Check: y = 2: 5² = 25; 49 − 28 + 4 = 25 ✅ (galat wale 45 aur 17 dete)
Doosri common galti: (a − b)² aur a² − b² ko ek samajhna. 5² − 3² = 16, par (5 − 3)² = 4.
🧠 Rule of thumb: (−b)² hamesha +b²: (a − b)² ka aakhri term plus; sirf beech wala minus.
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📝 10 MCQs
Q1. (a − b)² = ?
Solution dekho
Sahi jawab: (b)
- (a) galat: Yeh to (a + b)(a − b) hai.
- (b) sahi: Kona wapas joda: + b² ✅
- (c) galat: (−b)² = +b², minus nahi.
- (d) galat: Beech ka −2ab bhool gaye.
Q2. 79² = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 6400 − 160 + 1 = 6241 ✅
- (b) galat: Yeh 80² hai, 79² nahi.
- (c) galat: +1 ki jagah −79 kar diya.
- (d) galat: Ghataav ka hisaab galat.
Q3. 56 × 64 = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 60² − 4² = 3600 − 16 ✅
- (b) galat: 16 ghataana bhool gaye.
- (c) galat: Ghataav galat kiya, isliye yeh galat.
- (d) galat: Ghataana tha, jodna nahi.
Q4. (b − a)² vs (a − b)²?
Solution dekho
Sahi jawab: (a)
- (a) sahi: −(a − b) ka square same ✅
- (b) galat: Square mein sign gayab hota.
- (c) galat: Square mein sign gayab hota.
- (d) galat: Algebra se saaf pata chalta.
Q5. 12² − 8² = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: (12 + 8)(12 − 8) = 20 × 4 ✅
- (b) galat: 16 to (12 − 8)² hai, isliye yeh galat.
- (c) galat: 4 to sirf farak hai, isliye yeh galat.
- (d) galat: 208 to jod hai, farak nahi.
Q6. Sridhara se 52²?
Solution dekho
Sahi jawab: (a)
- (a) sahi: (52 + 2)(52 − 2) + 2² ✅
- (b) galat: Yeh identity nahi banti.
- (c) galat: 2ab bhool gaye, isliye yeh galat.
- (d) galat: b² jodna tha, ghataana nahi.
Q7. (3x − 4)² = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: (3x)² − 2·3x·4 + 16 ✅
- (b) galat: Beech ka −24x bhool gaye.
- (c) galat: 2 × 3x × 4 = 24x hota.
- (d) galat: (3x)² = 9x², 3x² nahi.
Q8. 2(a² + b²) = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 2ab aur −2ab kat jaate ✅
- (b) galat: Isme 2ab bachta hai, isliye yeh galat.
- (c) galat: Isme −2ab bachta hai, isliye yeh galat.
- (d) galat: Yeh to farak hai, isliye yeh galat.
Q9. 15 ko a² − b² mein (positive integers)?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 15 × 1 aur 5 × 3 se ✅
- (b) galat: Odd number hamesha ban jaata.
- (c) galat: 10 square nahi hai, isliye yeh galat.
- (d) galat: 8² − 7² bhi chalta hai.
Q10. Frame 20 cm, photo 16 cm: frame area?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 36 × 4 = 144 ✅
- (b) galat: 16 to (20 − 16)² hai, isliye yeh galat.
- (c) galat: 400 to poora square hai.
- (d) galat: 256 to sirf photo hai.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. (a − b)² ke liye alag formula yaad karna padega?
Nahi: (a + b)² mein b ki jagah −b daalo. Bas.
2. a² − b² ka chitra?
a² mein se kone ka b² hatao; bache L ko kaat ke (a + b) × (a − b) rectangle banao.
3. Kaunse numbers do squares ka farak ban sakte?
Har odd number aur har 4 ka multiple. 2, 6, 10 (4k + 2) nahi.
4. Kya fractions pe bhi?
Haan, identities har number pe: (1/2 + 1/3)² bhi.
5. Sridharacharya kaun?
Lagbhag 750 CE ke Bharatiya ganitagya; Patiganita aur Trishatika ke lekhak.
6. Aage kya?
Algebra ki galtiyaan pakadna aur sudhaarna: Post 6 — Algebra Ki Galti Pakdo.
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy (x − 3)² kholo.
Solution dekho
x² − 6x + 9. Check: x = 5 → 4 ✅
Q2. Easy 39²?
Solution dekho
1600 − 80 + 1 = 1521. Check ✅
Q3. Easy 23 × 17?
Solution dekho
400 − 9 = 391. Check ✅
Q4. Easy (y + 6)(y − 6)?
Solution dekho
y² − 36. Check: y = 7 → 13 ✅
Q5. Medium (4a − 3b)² kholo.
Solution dekho
16a² − 24ab + 9b². Check: a = b = 1 → 1 ✅
Q6. Medium 994²?
Solution dekho
1000000 − 12000 + 36 = 988036. Check ✅
Q7. Medium 1005 × 995?
Solution dekho
1000000 − 25 = 999975. Check ✅
Q8. Medium 29² − 21²?
Solution dekho
50 × 8 = 400. Check: 841 − 441 ✅
Q9. NCERT style Sridhara se 86²?
Solution dekho
90 × 82 + 16 = 7380 + 16 = 7396. Check ✅
Q10. NCERT style (2a − 1/2)² kholo.
Solution dekho
4a² − 2a + 1/4. Check: a = 1 → 9/4 ✅
Q11. NCERT style (x + y)² − (x − y)²?
Solution dekho
4xy. Check: x = 3, y = 1 → 16 − 4 ✅
Q12. NCERT style (6n + 1)² − (6n − 1)² hamesha kiska multiple?
Solution dekho
24n → 24. Check: n = 1 → 49 − 25 = 24 ✅
Answer Key — charumam.com exclusive
🧠 HOTS (Sochne Wale Sawaal)
Q1. Teen lagaataar numbers: beech ka square − baaki do ka guna?
Solution dekho
n² − (n − 1)(n + 1) = 1 hamesha. Check: 6² − 5 × 7 = 1 ✅
Q2. Kya 30 do squares ka farak?
Solution dekho
30 = 4k + 2 → nahi: (a + b)(a − b) mein dono factor same parity, to guna odd ya 4 ka multiple. Check ✅
Q3. (a − b)(a + b)(a² + b²)?
Solution dekho
a⁴ − b⁴. Check: a = 2, b = 1 → 15 ✅
Q4. Square park 50 m, andar chaaron taraf 2 m raasta. Raasta ka area?
Solution dekho
50² − 46² = 96 × 4 = 384 m². Check ✅
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🏆 Olympiad Corner
Q1. 2025² − 2024²?
Solution dekho
4049.
Q2. 101² − 99²?
Solution dekho
400.
Q3. a − b = 3, ab = 10: a² + b²?
Solution dekho
9 + 20 = 29.
Q4. 1 se 100 mein kitne numbers do squares (0² bhi chalega) ka farak nahi ban sakte?
Solution dekho
4k + 2 wale → 25.
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💗 CharuMam Motivation
Iqbal chacha ka kone ka tukda aapke dimaag mein reh gaya to (a − b)² kabhi galat nahi hoga. Chitra = saboot = yaad! ✂️
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🟢 Master Recap
- 💡 Chaabi: a² mein se do pattiyaan (a × b) kaato to kone ka b × b tukda do baar kat jaata, isliye ek baar wapas jodo: (a − b)² = a² − 2ab + b². Aur jod × farak mein pattiyaan kat jaati: (a + b)(a − b) = a² − b².
- (a − b)² = a² − 2ab + b².
- Kona do baar kata → ek baar wapas.
- (a + b)(a − b) = a² − b².
- Sridhara: a² = (a + b)(a − b) + b².
- (a + b)² + (a − b)² = 2(a² + b²).
- (a − b)² = (b − a)².
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👉 Aage Kya?
Algebra ki aam galtiyaan pakdo aur sudhaaro: Post 6 — Algebra Ki Galti Pakdo.
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