1. Algebraic Identities Kya Hain? 🧩
Algebraic Identity ek aisi equation hoti hai jo variables ki har value ke liye true hoti hai. Ye math ke "shortcuts" ya "formulas" hain jo humein lambi calculations se bachate hain.
Imagine karo aap ek video game khel rahe ho. Ek level bohot mushkil hai, jisme dher saare enemies (Complex Calculations) hain.
Lekin aapke paas ek secret "Cheat Code" (Identity) hai! Jaise hi aap code enter karte ho, saare enemies gayab ho jate hain aur aap seedha level paar kar lete ho.
Algebraic Identities wahi cheat codes hain. Instead of multiplying (x + 3)(x + 3)
step-by-step, aap seedha formula lagate ho aur answer mil jata hai! 🚀
🎥 Video Explanation
English Explanation
Hindi Explanation
📊 Infographic Summary
2. Standard Identities (The Basics) 🛠️
Ye wo 4 basic identities hain jo aapne pehle bhi dekhi hongi. Inhe yaad rakhna bohot zaroori hai!
Example 1: Find 105 × 106 without multiplying directly.
Solution:
Hum isse Identity IV (x + a)(x + b) use karke solve kar sakte hain.
105 × 106 = (100 + 5)(100 + 6)
Yahan x = 100, a = 5, b = 6
Formula: x² + (a + b)x + ab
= (100)² + (5 + 6)(100) + (5 × 6)
= 10000 + 1100 + 30
= 11130
Dekha? Bina lambi multiplication ke answer aa gaya! ✨
Example 2: Evaluate (99)² using identities.
Solution:
Hum 99 ko (100 - 1) likh sakte hain. Ab Identity II (x - y)² use karenge.
(99)² = (100 - 1)²
= (100)² - 2(100)(1) + (1)²
= 10000 - 200 + 1
= 9801
Example 3: Factorise 49a² - 25b²
Solution:
Ye x² - y² form mein hai. Hum isse (7a)² - (5b)² likh sakte hain.
Using Identity III: x² - y² = (x + y)(x - y)
= (7a + 5b)(7a - 5b)
3. Advanced Identities (Level Up!) 🚀
Ab thoda advance level par chalte hain. Ye identities cubic polynomials aur 3 variables ke liye hain.
Identity V: Square of a Trinomial
Identity VI & VII: Cubic Identities
Identity VIII: The Big One
Example 1: Expand (3a + 4b)³
Solution:
Using Identity VI: (x + y)³ = x³ + y³ + 3xy(x + y)
Yahan x = 3a aur y = 4b
= (3a)³ + (4b)³ + 3(3a)(4b)(3a + 4b)
= 27a³ + 64b³ + 36ab(3a + 4b)
= 27a³ + 64b³ + 108a²b + 144ab²
Example 2: Expand (4a - 2b - 3c)²
Solution:
Using Identity V: (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
Yahan x = 4a, y = -2b, z = -3c
= (4a)² + (-2b)² + (-3c)² + 2(4a)(-2b) + 2(-2b)(-3c) + 2(-3c)(4a)
= 16a² + 4b² + 9c² - 16ab + 12bc - 24ca
Example 3: Factorise 8x³ + 27y³ + 36x²y + 54xy²
Solution:
Isse hum (2x)³ + (3y)³ + 3(2x)²(3y) + 3(2x)(3y)² likh sakte hain.
Ye Identity VI (x + y)³ ka expanded form hai.
= (2x + 3y)³
= (2x + 3y)(2x + 3y)(2x + 3y)
🎉 Conclusion
Algebraic Identities math ke powerful tools hain. Inhe practice karke aap complex problems ko chutkiyon mein solve kar sakte hain. Agle chapter mein hum Coordinate Geometry ke baare mein seekhenge!