Algebraic Identities with Derivations | Important Formulas for Class 9 & 10 Students

Algebraic identities make solving maths problems easier, faster, and more accurate. They help in simplifying expressions, factorization, and solving equations in exams. These formulas are the foundation for higher classes and are very useful in competitive exams like JEE and NEET. By mastering them, students build strong logical thinking and problem-solving skills that are valuable for future studies and careers.

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All Algebraic Identities and Their Derivations

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1. (a + b)² = a² + 2ab + b²

Derivation: (a + b)² = (a + b)(a + b)
= a(a + b) + b(a + b)
= a² + ab + ab + b²
= a² + 2ab + b²

2. (a – b)² = a² – 2ab + b²

Derivation: (a – b)² = (a – b)(a – b)
= a(a – b) – b(a – b)
= a² – ab – ab + b²
= a² – 2ab + b²

3. (a + b)(a – b) = a² – b²

Derivation: (a + b)(a – b) = a(a – b) + b(a – b)
= a² – ab + ab – b²
= a² – b²

4. (x + a)(x + b) = x² + (a + b)x + ab

Derivation: (x + a)(x + b) = x(x + b) + a(x + b)
= x² + xb + ax + ab
= x² + (a + b)x + ab

5. (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

Derivation: (a + b + c)² = (a + b + c)(a + b + c)
= a² + b² + c² + ab + ba + bc + cb + ca + ac
= a² + b² + c² + 2ab + 2bc + 2ca

6. (x + a)³ = x³ + 3x²a + 3xa² + a³

Derivation: (x + a)³ = (x + a)(x + a)(x + a)
= (x² + 2xa + a²)(x + a)
= x³ + x²a + 2x²a + 2xa² + xa² + a³
= x³ + 3x²a + 3xa² + a³

7. (x – a)³ = x³ – 3x²a + 3xa² – a³

Derivation: (x – a)³ = (x – a)(x – a)(x – a)
= (x² – 2xa + a²)(x – a)
= x³ – 3x²a + 3xa² – a³

8. x³ + y³ = (x + y)(x² – xy + y²)

Derivation: (x + y)(x² – xy + y²) = x³ – x²y + xy² + x²y – xy² + y³
= x³ + y³

9. x³ – y³ = (x – y)(x² + xy + y²)

Derivation: (x – y)(x² + xy + y²) = x³ + x²y + xy² – x²y – xy² – y³
= x³ – y³

Solved Examples of Algebraic Identities

.example { font-size: 18px; margin: 20px 0; padding: 12px 0; border-bottom: 1px solid #eee; } .example h3 { font-size: 16px; margin: 0 0 10px 0; color: #0a3d7a; background: #e9f4ff; /* BG only for the identity/problem line */ display: inline-block; padding: 6px 12px; border-radius: 6px; } .derivation { text-align: leftr; line-height: 1.8; font-size: 14px; margin-top: 8px; } .derivation b { display: block; color: #c91919; margin-bottom: 6px; } .final { font-weight: 700; }

Example 1: Expand (2x + 3y)²

Solution: Use (a + b)² = a² + 2ab + b²
a = 2x, b = 3y
(2x + 3y)² = (2x)² + 2(2x)(3y) + (3y)²
= 4x² + 12xy + 9y²
Answer: 4x² + 12xy + 9y²

Example 2: Expand (5a − 2b)²

Solution: Use (a − b)² = a² − 2ab + b²
a = 5a, b = 2b
(5a − 2b)² = (5a)² − 2(5a)(2b) + (2b)²
= 25a² − 20ab + 4b²
Answer: 25a² − 20ab + 4b²

Example 3: Expand (3p + 4)(3p − 4)

Solution: Use (a + b)(a − b) = a² − b²
a = 3p, b = 4
(3p + 4)(3p − 4) = (3p)² − 4² = 9p² − 16
Answer: 9p² − 16

Example 4: Expand (x + 5)(x + 7)

Solution: Use (x + a)(x + b) = x² + (a + b)x + ab
a = 5, b = 7
(x + 5)(x + 7) = x² + 12x + 35
Answer: x² + 12x + 35

Example 5: Expand (x + 2y − 3z)²

Solution: Use (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
a = x, b = 2y, c = −3z
= x² + (2y)² + (−3z)² + 2(x)(2y) + 2(2y)(−3z) + 2(−3z)(x)
= x² + 4y² + 9z² + 4xy − 12yz − 6xz
Answer: x² + 4y² + 9z² + 4xy − 12yz − 6xz

Example 6: Evaluate 99² using identities

Solution: 99 = 100 − 1 → use (a − b)²
99² = 100² − 2·100·1 + 1² = 10000 − 200 + 1
= 9801
Answer: 9801

Example 7: Evaluate 51 × 49 using identities

Solution: (50 + 1)(50 − 1) = 50² − 1² (use a² − b²)
= 2500 − 1 = 2499
Answer: 2499

Example 8: Expand (2x − 3)³

Solution: Use (a − b)³ = a³ − 3a²b + 3ab² − b³
a = 2x, b = 3
(2x − 3)³ = (2x)³ − 3(2x)²(3) + 3(2x)(3)² − 3³
= 8x³ − 36x² + 54x − 27
Answer: 8x³ − 36x² + 54x − 27

Example 9: Factor x³ + 8

Solution: Use a³ + b³ = (a + b)(a² − ab + b²)
a = x, b = 2
x³ + 8 = (x + 2)(x² − 2x + 4)
Answer: (x + 2)(x² − 2x + 4)

Example 10: Factor 27y³ − 1

Solution: Use a³ − b³ = (a − b)(a² + ab + b²)
a = 3y, b = 1
27y³ − 1 = (3y − 1)\,(9y² + 3y + 1)
Answer: (3y − 1)(9y² + 3y + 1)

MCQ Practice Questions on Algebraic Identities

1. The expansion of (a+b)² is:

a² + b²
a² + 2ab + b²
a² – 2ab + b²
a² – b²

See Answer

✅ Correct Answer: B) a² + 2ab + b²

See Explanation

📘 (a+b)² = (a+b)(a+b) = a² + 2ab + b².

2. The value of (x+y)(x−y) is:

x² + y²
x² – y²
x² + 2xy + y²
x² – 2xy + y²

See Answer

✅ Correct Answer: B) x² – y²

See Explanation

📘 (x+y)(x−y) = x² – y² (difference of squares).

3. The square of 2a is:

2a²
a²
4a²
8a²

See Answer

✅ Correct Answer: C) 4a²

See Explanation

📘 (2a)² = 2a × 2a = 4a².

4. The expansion of (a−b)² is:

a² + b²
a² – 2ab + b²
a² + 2ab + b²
a² – b²

See Answer

✅ Correct Answer: B) a² – 2ab + b²

See Explanation

📘 (a−b)² = a² – 2ab + b².

5. The expansion of (x+3)² is:

x² + 6x + 9
x² – 6x + 9
x² + 3x + 9
x² – 3x + 9

See Answer

✅ Correct Answer: A) x² + 6x + 9

See Explanation

📘 (x+3)² = x² + 2×x×3 + 3² = x² + 6x + 9.

6. The expansion of (2x+5)² is:

4x² + 20x + 25
2x² + 25
4x² + 5
x² + 10x + 25

See Answer

✅ Correct Answer: A) 4x² + 20x + 25

See Explanation

📘 (2x+5)² = 4x² + 20x + 25.

7. The cube of (a+b) is:

a³ + b³
a³ + 3a²b + 3ab² + b³
a³ + 3ab + b³
a³ – 3a²b + b³

See Answer

✅ Correct Answer: B) a³ + 3a²b + 3ab² + b³

See Explanation

📘 (a+b)³ = a³ + 3a²b + 3ab² + b³.

8. The expansion of (x−2)² is:

x² + 4x + 4
x² – 4x + 4
x² + 2x + 4
x² – 2x + 4

See Answer

✅ Correct Answer: B) x² – 4x + 4

See Explanation

📘 (x−2)² = x² – 4x + 4.

9. The value of (x+1)(x+2) is:

x² + 2x + 1
x² + 3x + 2
x² + x + 2
x² + 4x + 4

See Answer

✅ Correct Answer: B) x² + 3x + 2

See Explanation

📘 (x+1)(x+2) = x² + 3x + 2.

10. The expansion of (3x+4)² is:

9x² + 24x + 16
6x² + 16
9x² + 12x + 16
x² + 7x + 16

See Answer

✅ Correct Answer: A) 9x² + 24x + 16

See Explanation

📘 (3x+4)² = 9x² + 24x + 16.

11. The expansion of (a+b+c)² is:

a²+b²+c²
a²+b²+c²+2ab+2bc+2ca
a²+b²+c²+ab+bc+ca
a²+b²+c²-2ab-2bc-2ca

See Answer

✅ Correct Answer: B) a²+b²+c²+2ab+2bc+2ca

See Explanation

📘 (a+b+c)² = a²+b²+c²+2ab+2bc+2ca.

12. The value of (x−3)(x+3) is:

x² – 9
x² + 9
x² – 6x + 9
x² + 6x + 9

See Answer

✅ Correct Answer: A) x² – 9

See Explanation

📘 (x−3)(x+3) = x² – 9.

13. The expansion of (5x−1)² is:

25x² – 10x + 1
25x² + 10x + 1
5x² – 2x + 1
25x² – 1

See Answer

✅ Correct Answer: A) 25x² – 10x + 1

See Explanation

📘 (5x−1)² = 25x² – 10x + 1.

14. The cube of (x+2) is:

x³ + 6x² + 12x + 8
x³ + 3x² + 6x + 8
x³ + 12x + 8
x³ + 2x + 8

See Answer

✅ Correct Answer: A) x³ + 6x² + 12x + 8

See Explanation

📘 (x+2)³ = x³ + 6x² + 12x + 8.

15. The expansion of (a+b)(a−b) is:

a²+b²
a²−b²
a²+2ab+b²
a²−2ab+b²

See Answer

✅ Correct Answer: B) a²−b²

See Explanation

📘 (a+b)(a−b) = a²−b².

16. The value of (2x+3)(2x−3) is:

4x² – 9
4x² + 9
2x² – 6
2x² + 9

See Answer

✅ Correct Answer: A) 4x² – 9

See Explanation

📘 (2x+3)(2x−3) = 4x² – 9.

17. The expansion of (x+4)² is:

x² + 4x + 16
x² + 8x + 16
x² + 2x + 16
x² + 16

See Answer

✅ Correct Answer: B) x² + 8x + 16

See Explanation

📘 (x+4)² = x² + 8x + 16.

18. The cube of (a−b) is:

a³ – 3a²b + 3ab² – b³
a³ + 3a²b – 3ab² + b³
a³ – b³
a³ + b³

See Answer

✅ Correct Answer: A) a³ – 3a²b + 3ab² – b³

See Explanation

📘 (a−b)³ = a³ – 3a²b + 3ab² – b³.

19. The value of (x+5)(x+2) is:

x² + 7x + 10
x² + 5x + 10
x² + 2x + 5
x² + 10

See Answer

✅ Correct Answer: A) x² + 7x + 10

See Explanation

📘 (x+5)(x+2) = x² + 7x + 10.

20. If x + y = 5 and xy = 6, what is x² + y²?

A) 19
B) 13
C) 25
D) 31

See Answer

✅ Correct Answer: A) 19

See Explanation

📘 x² + y² = (x + y)² – 2xy = 5² – 2×6 = 25 – 12 = 13.
Correction: The correct result is 13, not 19. ✅

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Anonymous

September 10, 2025

Kitna kama liye sir blogging krke mujhe bhi sikhna hai sir sikha dijiye

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