Algebra Ki Galti Pakdo | Class 8 Maths Book 1 Ch 6

📘 Class 8 · Ganita Prakash Book 1Ch 6 · We Distribute, Yet Things MultiplyPost 6 / 7⏱ 25–30 min

Algebra Ki Galti Pakdo 🔎

IS POST MEIN

Pehle Yeh Aata Hai?Kahani Concept Deep Concept Examples Real Life Galti 10 MCQs FAQs Practice HOTS Olympiad Recap

✅ Pehle Yeh Aata Hai?

Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:

3x + 4x = ?
3x + 4y = ?
(a + b)² = ?

Answers dekho

7x
3x + 4y (aage nahi jud sakte)
a² + 2ab + b²

Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 1 — Distributive Property Kya Hai? se Post 5 — (a − b)² Aur (a + b)(a − b) tak ek baar dekh lijiye.

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🎬 Kahani: Teacher Ki Laal Pen Aur Golu Ka Sawaal

Golu ki copy mein likha tha: (2a + 3)² = 4a² + 9. Teacher ne laal gola lagaaya. Golu: “Ma’am, a = 0 daal ke dekha tha: 9 = 9! Sahi hai na?” 🖍️

Ma’am: “Ab a = 1 daalo.” Golu: 5² = 25… aur 4 + 9 = 13. 😳

Ek value pe barabar hona saboot nahi, par ek value pe na hona galti ka pakka saboot! Aaj galtiyaan pakadne ka jasoosi kit banate hain. 🕵️

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📖 Concept: Identity Har Value Pe Sach; Ek Bhi Value Pe Jhooth = Galti

💡 Is post ki ek badi baat: Identity ka matlab: letters ki har value pe dono taraf barabar. To ek chhoti value (1, 2, ya −1) daalo: barabar nahi aaya to galti pakki. Phir dekho kaun sa niyam toota: distributive, sign, like terms ya 2ab.

Step 1: Jasoosi kit: 4 test

Value test: x = 2 jaisa daalo (0 aur 1 se bacho, woh dhokha de sakte).
Tukde gino: 2-term × 2-term = 4 tukde. Kam dikhe? Kuch chhoota.
Sign test: har minus ke saath guna dobara dekho.
Like terms: sirf bilkul same letter-power wale jodo.

WHY 0, 1 se bacho? 0 pe kai terms gayab, 1 pe x² = x: galti chhup jaati (Golu ka a = 0!).

Step 2: Galti 1: adhoora distributive

❌ 4(x − 2) + 2(x + 3) = 4x − 2 + 2x + 3 = 6x + 1

✅ 4x − 8 + 2x + 6 = 6x − 2. Check: x = 2: 0 + 10 = 10; 6 × 2 − 2 = 10 ✅ (galat 13)

❌ (1/3)(9y − 6) + 2 = 3y − 6 + 2 → ✅ 3y − 2 + 2 = 3y.

Step 3: Galti 2: gayab 2ab ya gayab tukde

❌ (2a + 3)² = 4a² + 9 → ✅ 4a² + 12a + 9.

❌ (x + 3)(x + 4) = x² + 12 → ✅ x² + 7x + 12. Check: x = 2: 5 × 6 = 30; 4 + 14 + 12 = 30 ✅

Step 4: Galti 3: sign aur power

❌ −2m(3m − n) = −6m − 2mn

✅ −2m × 3m = −6m² (power!), −2m × (−n) = +2mn → −6m² + 2mn. Check: m = 2, n = 1: −4 × 5 = −20; −24 + 4 = −20 ✅

Step 5: Galti 4: unlike terms jodna, guna pe baantna

❌ 3x² + 4x = 7x³. ✅ Aage saral nahi hota: 3x² + 4x (ya x(3x + 4)).

❌ 2p(3q × 4r) = 6pq × 8pr. ✅ Guna guna pe nahi baant-ta: 2p × 12qr = 24pqr.

❌ x²y + xy² + xy = xy(x + y). ✅ xy(x + y + 1) (aakhri term ka 1!).

Step 6: Jaal: jo galat lagta par sahi hai

(−3 + k)² = k² − 6k + 9: sahi! Kyunki (−3 + k) = (k − 3).

Sabak: galti pakadne se pehle bhi value test. k = 5: 2² = 4; 25 − 30 + 9 = 4 ✅

Kab kaunsa test? Value test sabse tez (har galti pakadta jo value pe dikhe). Tukde/sign test batata kahan galti hai, taaki sudhaar sako.

🪜 Atak Gaye? Aise Socho

Line-by-line padho, ek baar mein ek step.
Har step pe x = 2 daal ke dekho: kahaan toota?
Toote step mein: distributive poora? sign sahi? power sahi?
Sahi step likho.
Aakhir mein poore jawab ko phir value test.

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🔐 Deep Concept: Galti Bhi Ek Pattern Hai

Algebra ki zyaadatar galtiyaan ek hi “gande niyam” se aati: “jo cheez jod pe baant-ti hai, woh sab pe baant-ti hogi.”

Guna jod pe baant-ta ✅ → par square jod pe nahi: (a + b)² ≠ a² + b².
Guna jod pe ✅ → par guna guna pe nahi: 2(3 × 4) ≠ 6 × 8.
√ bhi jod pe nahi: √(9 + 16) = 5, par 3 + 4 = 7.

Mathematician kaise sochte? Har naye niyam ke saath poochho: “Yeh kis operation pe baant-ta hai, kis pe nahi?” Aur shak ho to value test.

Unifying idea: identity = har value pe sach. Isliye value test sabse bada jasoos, aur chitra (area) sabse bada saboot. 🎯

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✏️ Solved Examples

Example 1 (Easy): Galti dhundho: 3(2x + 5) − 2(x − 4) = 6x + 15 − 2x − 8 = 4x + 7.

−2 × (−4) = +8, −8 nahi. Sahi: 6x + 15 − 2x + 8 = 4x + 23.

Check: x = 2: 3 × 9 − 2 × (−2) = 27 + 4 = 31; 8 + 23 = 31 ✅

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Example 2 (Medium): Galti dhundho: (y − 5)² = y² − 25.

Beech ka −10y aur sign: sahi y² − 10y + 25.

Check: y = 2: (−3)² = 9; 4 − 20 + 25 = 9 ✅ (galat −21)

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Example 3 (NCERT style): Galti dhundho: 5w² + 6w = 11w².

w² aur w unlike terms. Sahi: 5w² + 6w (ya w(5w + 6)).

Check: w = 2: 20 + 12 = 32; 11 × 4 = 44 ≠ 32 → galti pakki ✅

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Golu ka naya niyam: “Ab main 2 daal ke check karta hoon, 0 nahi!” Agli test mein usne apni do galtiyaan khud pakdi, teacher se pehle. 🏆

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🏠 Galti Pakadna Roz Ki Zindagi Mein

🧾 Bill: dukaandaar ke kul ko ek-do cheez ke anumaan se check karo.
💻 Programmer: code ko chhote input pe test karte (value test!).
✈️ Pilot checklist: har step pe check, taaki ek galti aage na badhe.
🏗️ Engineer: design ko alag tareeke se dobara hisaab karte.
🍳 Recipe double karna: namak bhi double? Har cheez ko guna karna (distributive!).

Recipe wali galti sab karte: 2 log ke liye recipe ko 4 log ke liye karo to har cheez double, sirf ek nahi. Wahi adhoora distributive!

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🧭 Kab Use Karein, Kab Nahi

Situation	Kya karein	Kyun
Shak hai ki galat	✅ x = 2 daalo	0, 1 se bacho
Kahaan galti?	✅ Har step pe value	Toota step pakdo
Brackets	✅ Har term ko guna	Sign ke saath
Square	✅ 2ab dhundho	Chaar tukde
Jodna	✅ Sirf like terms	Power same?

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❌→✅ Worked Mistake

Sawaal: Golu ne (a + 4)(a − 1) = a² + 3a − 4 likha aur a = 1 pe check kiya: 5 × 0 = 0; 1 + 3 − 4 = 0 ✅. Phir bhi teacher ne double-check bola. Sahi hai?

Dekho: a = 2: 6 × 1 = 6; 4 + 6 − 4 = 6 ✅. Haan, sahi hai. Par a = 1 akela kaafi nahi tha (us pe (a − 1) zero ho gaya, aadha expression gayab).

❌ Galti ka tareeka: sirf woh value daalna jo bracket ko 0 kare. ✅ Sahi tareeka: “aam” value (2, 3, −2) daalo. Check: a = −2: 2 × (−3) = −6; 4 − 6 − 4 = −6 ✅

🧠 Rule of thumb: Ek value pe jhooth = galti pakki; ek value pe sach = shayad. 2 ya 3 daal ke test karo.

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📝 10 MCQs

Q1. (3x + 2)² = 9x² + 4 mein galti?

(a) 2ab gayab

(b) Sign

(c) Power

(d) Koi nahi

Solution dekho

Sahi jawab: (a)

(a) sahi: 12x gayab hai, 2ab wala tukda ✅
(b) galat: Koi minus hai hi nahi yahan.
(c) galat: 9x² sahi power hai, isliye yeh galat.
(d) galat: x = 1 pe 25 ≠ 13, galti hai.

Q2. 2(x − 3) = 2x − 3 mein galti?

(a) Adhoora distributive

(b) Sign

(c) Square

(d) Like terms

Solution dekho

Sahi jawab: (a)

(a) sahi: 3 ko bhi 2 se guna: 2x − 6 ✅
(b) galat: Sign to theek hai, number galat.
(c) galat: Yahan koi square hai hi nahi.
(d) galat: Like terms ka sawaal nahi.

Q3. −3(a − 2) = −3a − 6 mein galti?

(a) Sign: +6 hona tha

(b) Koi nahi

(c) Power

(d) 2ab

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi jawab: −3 × −2 = +6 ✅
(b) galat: a = 2: 0 vs −12, galti hai.
(c) galat: Power ka koi sawaal nahi.
(d) galat: Square nahi hai yahan.

Q4. 4x + 3x² = 7x³?

(a) Unlike terms jode

(b) Sahi hai

(c) Sign

(d) Distributive

Solution dekho

Sahi jawab: (a)

(a) sahi: x aur x² alag terms hain ✅
(b) galat: x = 2 pe 20 vs 56, isliye yeh galat.
(c) galat: Sign ka koi sawaal nahi.
(d) galat: Bracket hai hi nahi yahan.

Q5. Value test ke liye sabse achhi value?

(a) 2

(b) 0

(c) 1

(d) 100

Solution dekho

Sahi jawab: (a)

(a) sahi: 2 pe galti chhupti nahi ✅
(b) galat: 0 pe terms gayab ho jaate.
(c) galat: 1 pe x² = x, dhokha, isliye yeh galat.
(d) galat: Bada number hisaab mushkil.

Q6. (−4 + m)² = m² − 8m + 16?

(a) Sahi hai

(b) Galat sign

(c) 2ab gayab

(d) Galat power

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi jawab: (m − 4)² hi to hai ✅
(b) galat: Sign bilkul sahi hai, isliye yeh galat.
(c) galat: −8m hi 2ab hai, isliye yeh galat.
(d) galat: m² sahi power hai, isliye yeh galat.

Q7. 2a(3b × 5c) = ?

(a) 30abc

(b) 6ab × 10ac

(c) 6ab + 10ac

(d) 11abc

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi jawab: 2a × 15bc = 30abc ✅
(b) galat: Guna guna pe nahi baant-ta.
(c) galat: Bracket mein jod nahi, guna hai.
(d) galat: Guna ki jagah jod kar diya.

Q8. a²b + ab² + ab = ab(?)

(a) a + b + 1

(b) a + b

(c) ab + 1

(d) a + b + ab

Solution dekho

Sahi jawab: (a)

(a) sahi: ab ÷ ab = 1 bhi likho ✅
(b) galat: Aakhri term ka 1 bhool gaye.
(c) galat: Galat tarah se common nikala.
(d) galat: ab ÷ ab = 1 hota, ab nahi.

Q9. (x + 2)(x + 5) = x² + 10?

(a) Beech ke tukde gayab

(b) Sahi

(c) Sign

(d) Power

Solution dekho

Sahi jawab: (a)

(a) sahi: 7x gayab: x² + 7x + 10 ✅
(b) galat: x = 1 pe 18 vs 11, isliye yeh galat.
(c) galat: Sign ka koi sawaal nahi.
(d) galat: Power to theek hai, isliye yeh galat.

Q10. √(16 + 9) = 4 + 3?

(a) Nahi, 5 ≠ 7

(b) Haan

(c) Kabhi-kabhi

(d) Sirf integers

Solution dekho

Sahi jawab: (a)

(a) sahi: √ jod pe nahi baant-ta ✅
(b) galat: 5 aur 7 alag hain, isliye yeh galat.
(c) galat: Yahan to bilkul nahi, isliye yeh galat.
(d) galat: Integers mein bhi nahi.

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❓ FAQs (Bachchon Ke Asli Sawaal)

1. Value test kaafi hai?

Galti pakadne ke liye haan. Sahi saabit karne ke liye nahi: uske liye algebra.

2. 0 aur 1 kyun nahi?

0 pe bahut terms gayab, 1 pe x² = x³ = x. Galti chhup jaati.

3. Like terms kaise pehchaanein?

Same letters, same power: 3x²y aur 5x²y like; 3x²y aur 5xy² nahi.

4. Sabse aam galti?

(a + b)² = a² + b², aur bracket ke aage ke minus ko andar na le jaana.

5. Kya calculator se check kar sakte?

Haan: dono taraf ek value daal ke. Yahi value test hai.

6. Aage kya?

Ek sawaal ke kai sahi tareeke: Post 7 — Ek Sawaal, Kai Tareeke.

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🏋️ Practice Zone (12 Sawaal)

Q1. Easy Galti: 5(y + 2) = 5y + 2.

Solution dekho

Sahi 5y + 10. Check: y = 2 → 20 ✅

Q2. Easy Galti: (m + 1)² = m² + 1.

Solution dekho

Sahi m² + 2m + 1. Check: m = 2 → 9 ✅

Q3. Easy Galti: 4a + 5b = 9ab.

Solution dekho

Unlike: 4a + 5b. Check: a = b = 2 → 18 vs 36 ✅

Q4. Easy Galti: −(x − 7) = −x − 7.

Solution dekho

Sahi −x + 7. Check: x = 2 → 5 ✅

Q5. Medium Galti: 3p(p + 4q) = 3p + 12pq.

Solution dekho

Sahi 3p² + 12pq. Check: p = 2, q = 1 → 36 ✅

Q6. Medium Galti: (2k − 3)(k + 1) = 2k² − 3.

Solution dekho

Sahi 2k² − k − 3. Check: k = 2 → 3 ✅

Q7. Medium Galti: 6 + 3(x + 2) = 9(x + 2).

Solution dekho

Pehle guna: 6 + 3x + 6 = 3x + 12. Check: x = 2 → 18 vs 36 ✅

Q8. Medium Sahi ya galat: (−a − b)² = a² + 2ab + b².

Solution dekho

Sahi: −(a + b) ka square. Check: a = 1, b = 2 → 9 ✅

Q9. NCERT style Galti: (1/2)(8t − 4) − 2 = 4t − 4 − 2 = 4t − 6.

Solution dekho

Sahi: 4t − 2 − 2 = 4t − 4. Check: t = 2 → 4 ✅

Q10. NCERT style Galti: 2a² + 3a² + 4a²b = 9a⁴b.

Solution dekho

Sahi: 5a² + 4a²b. Check: a = 2, b = 1 → 36 ✅

Q11. NCERT style Galti: (x + y)(x − y) = x² − 2xy − y².

Solution dekho

Sahi x² − y². Check: x = 3, y = 1 → 8 ✅

Q12. NCERT style Galti: (3n)² − (2n)² = n².

Solution dekho

9n² − 4n² = 5n². Check: n = 2 → 20 ✅

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🧠 HOTS (Sochne Wale Sawaal)

Q1. Ek expression x = 1, 2, 3 teeno pe sahi aaya. Kya pakka identity?

Solution dekho

Nahi: x(x − 1)(x − 2)(x − 3) = 0 sirf in pe sach. Check: x = 4 → 24 ≠ 0 ✅

Q2. Kab a² + b² = (a + b)² sach?

Solution dekho

Jab 2ab = 0 → a = 0 ya b = 0. Check ✅

Q3. Riya: “(a + b)³ = a³ + b³.” Counter-example?

Solution dekho

a = b = 1: 8 ≠ 2. Sahi: a³ + 3a²b + 3ab² + b³. Check ✅

Q4. Kya (a/b + c/d) = (a + c)/(b + d)?

Solution dekho

Nahi: 1/2 + 1/2 = 1, par 2/4 = 1/2. Check ✅

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🏆 Olympiad Corner

Q1. (x + 1)² − (x − 1)² = 4: kab?

Solution dekho

4x = 4 → x = 1 (identity nahi, equation).

Q2. (a + b)² = a² + b² + 12, a = 2: b?

Solution dekho

4b = 12 → 3.

Q3. Galat: √(a² + b²) = a + b. Kab sach?

Solution dekho

Jab ab = 0 (a, b ≥ 0).

Q4. 3(2x − 1) − 2(3x − 2) = ?

Solution dekho

1 (x gayab!).

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💗 CharuMam Motivation

Galti karna sharm ki baat nahi; galti pakadna seekhna superpower hai. Golu jaise, apna value test har baar lagao! 🔎

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🟢 Master Recap

💡 Chaabi: Identity ka matlab: letters ki har value pe dono taraf barabar. To ek chhoti value (1, 2, ya −1) daalo: barabar nahi aaya to galti pakki. Phir dekho kaun sa niyam toota: distributive, sign, like terms ya 2ab.
Identity: har value pe sach.
Value test: 2 ya 3 daalo (0, 1 nahi).
Bracket ke har term ko guna.
(a ± b)² mein 2ab mat bhoolo.
Sirf like terms jodo.
Guna guna pe, square jod pe nahi baant-ta.

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👉 Aage Kya?

Ek hi pattern ko alag-alag tareekon se ginna, aur sab ka same aana: Post 7 — Ek Sawaal, Kai Tareeke.

← Pichla: Post 5 — (a − b)² Aur (a + b)(a − b)📚 Chapter 6 ke saare posts Agla: Post 7 — Ek Sawaal, Kai Tareeke →

Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬

✅ Pehle Yeh Aata Hai?

🎬 Kahani: Teacher Ki Laal Pen Aur Golu Ka Sawaal

📖 Concept: Identity Har Value Pe Sach; Ek Bhi Value Pe Jhooth = Galti

Step 1: Jasoosi kit: 4 test

Step 2: Galti 1: adhoora distributive

Step 3: Galti 2: gayab 2ab ya gayab tukde

Step 4: Galti 3: sign aur power

Step 5: Galti 4: unlike terms jodna, guna pe baantna

Step 6: Jaal: jo galat lagta par sahi hai

🪜 Atak Gaye? Aise Socho

🔐 Deep Concept: Galti Bhi Ek Pattern Hai

✏️ Solved Examples

🏠 Galti Pakadna Roz Ki Zindagi Mein

🧭 Kab Use Karein, Kab Nahi

❌→✅ Worked Mistake

📝 10 MCQs

❓ FAQs (Bachchon Ke Asli Sawaal)

🏋️ Practice Zone (12 Sawaal)

🧠 HOTS (Sochne Wale Sawaal)

🏆 Olympiad Corner

💗 CharuMam Motivation

🟢 Master Recap

👉 Aage Kya?

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