NCERT Class 9 Maths Exercise 4.1 Solutions – Exploring Algebraic Identities
Short Introduction
Algebraic identities are powerful mathematical tools that help us simplify calculations and expand algebraic expressions quickly. In Exercise 4.1, students learn how to apply the identity:
(a + b)² = a² + 2ab + b²
to expand algebraic expressions and calculate squares of numbers efficiently. This exercise strengthens algebraic manipulation skills and improves problem-solving speed for school examinations and competitive tests.
Quick Information Box
| Topic | Details |
|---|---|
| Chapter | Exploring Algebraic Identities |
| Exercise | 4.1 |
| Class | 9 |
| Main Identity Used | (a + b)² = a² + 2ab + b² |
| Skill Developed | Expansion & Mental Calculation |
| Difficulty Level | Easy to Moderate |
Concepts Used (Topics Covered)
✅ Algebraic Identities
✅ Expansion of Binomials
✅ Squaring Algebraic Expressions
✅ Squaring Rational Expressions
✅ Mental Mathematics using Identities
Important Formula
Identity Used
(a + b)² = a² + 2ab + b²
Where:
- a² = square of first term
- b² = square of second term
- 2ab = twice the product of both terms
Question 1
Using the identity (a + b)² = a² + 2ab + b², expand the following.
(i) (7x + 4y)²
(ii) (7x/5 + 3y/2)²
(iii) (2.5p + 1.5q)²
(iv) (3s/4 + 8t)²
(v) (x + 1/2y)²
(vi) (1/x + 1/y)²
Solution
Question 2
Using the same identity, find the values of:
(i) (64)²
(ii) (105)²
(iii) (205)²
Solution
Final Answers Summary
Q1
(i) 49x² + 56xy + 16y²
(ii) 49x²/25 + 21xy/5 + 9y²/4
(iii) 6.25p² + 7.5pq + 2.25q²
(iv) 9s²/16 + 12st + 64t²
(v) x² + x/y + 1/(4y²)
(vi) 1/x² + 2/xy + 1/y²
Q2
(i) 4096
(ii) 11025
(iii) 42025
Common Mistakes
❌ Forgetting the middle term 2ab.
❌ Squaring only one variable.
❌ Incorrect multiplication of fractions.
❌ Writing (a+b)² = a²+b².
Remember:
(a+b)² ≠ a²+b²
Correct identity:
(a+b)² = a² + 2ab + b²
Exam Tips
⭐ Always identify a and b first.
⭐ Write identity before solving.
⭐ Simplify fractions carefully.
⭐ For numerical squares, choose nearby multiples of 10, 100, etc.
⭐ Show each step clearly to get full marks.
Practice MCQs
1. (x+5)² equals
A. x²+25
B. x²+10x+25
C. x²+5x+25
D. x²+15x+25
✅ Answer: B
2. (2a+3b)² equals
A. 4a²+12ab+9b²
B. 4a²+6ab+9b²
C. 2a²+12ab+3b²
D. 4a²+9b²
✅ Answer: A
3. 102² equals
A. 10404
B. 10204
C. 10004
D. 12004
✅ Answer: A
4. Middle term in (a+b)² is
A. ab
B. a+b
C. 2ab
D. b²
✅ Answer: C
FAQ Section
Q1. What is an algebraic identity?
An algebraic identity is an equation that remains true for all values of the variables.
Q2. Which identity is used in Exercise 4.1?
(a+b)² = a² + 2ab + b²
Q3. Why is the term 2ab important?
It represents twice the product of the two terms and completes the square expansion correctly.
Q4. Can identities be used for mental calculations?
Yes. They help calculate squares of numbers quickly.
Q5. Is (a+b)² equal to a²+b²?
No.
(a+b)² = a² + 2ab + b²
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