1. Chapter at a Glance
How can you find 103 × 97 or 46² in your head in a few seconds? The secret is the distributive property and a handful of algebraic identities that follow from it. In this chapter you will multiply algebraic expressions, discover the identities (a + b)², (a − b)², (a + b)(a − b) and (x + a)(x + b) using area models, and use them for fast mental calculation.
- The distributive property: a(b + c) = ab + ac.
- Multiplying a binomial by a binomial.
- Standard identities and their geometric (area) meaning.
- Using identities for mental arithmetic.
- Patterns that identities explain.
2. The Distributive Property
Multiplication distributes over addition and subtraction:
Examples: 7 × 104 = 7 × 100 + 7 × 4 = 728; 3x(2x − 5) = 6x² − 15x.
3. Multiplying Two Binomials
Use the distributive property twice: multiply each term of the first bracket by each term of the second.
Expand (2x + 3)(x − 4).
Solution: 2x·x + 2x·(−4) + 3·x + 3·(−4) = 2x² − 8x + 3x − 12 = 2x² − 5x − 12.
4. Standard Identities
An identity is an equality that is true for all values of the letters.
| Identity | Expansion | Example |
|---|---|---|
| (a + b)² | a² + 2ab + b² | (x + 5)² = x² + 10x + 25 |
| (a − b)² | a² − 2ab + b² | (2y − 3)² = 4y² − 12y + 9 |
| (a + b)(a − b) | a² − b² | (m + 6)(m − 6) = m² − 36 |
| (x + a)(x + b) | x² + (a + b)x + ab | (x + 2)(x + 7) = x² + 9x + 14 |
Why (a + b)² = a² + 2ab + b²
A square of side (a + b) splits into a square a², a square b² and two rectangles ab. So the area is a² + 2ab + b². A common mistake is to write (a + b)² = a² + b² — the two rectangles are forgotten.
5. Identities for Mental Maths
- 103² = (100 + 3)² = 10000 + 600 + 9 = 10,609.
- 98² = (100 − 2)² = 10000 − 400 + 4 = 9,604.
- 103 × 97 = (100 + 3)(100 − 3) = 10000 − 9 = 9,991.
- 52² − 48² = (52 + 48)(52 − 48) = 100 × 4 = 400.
- 46² = (50 − 4)² = 2500 − 400 + 16 = 2,116.
- 105 × 108 = (100 + 5)(100 + 8) = 10000 + 1300 + 40 = 11,340.
6. Patterns Explained
- Product of two numbers that differ by 2 is one less than the square of the number between them: 9 × 11 = 99 = 10² − 1, because (n − 1)(n + 1) = n² − 1.
- The difference of squares of consecutive numbers is their sum: 15² − 14² = 29 = 15 + 14.
- Squares of numbers ending in 5: (10a + 5)² = 100a(a + 1) + 25.
7. Solved Problems (NCERT Style)
If a + b = 9 and ab = 20, find a² + b².
Solution: a² + b² = (a + b)² − 2ab = 81 − 40 = 41.
Simplify (x + 4)² − (x − 4)².
Solution: (x² + 8x + 16) − (x² − 8x + 16) = 16x. (Or use difference of squares: (2x)(8) = 16x.)
A square garden of side x m has a path 2 m wide around it outside. Find the area of the path.
Solution: (x + 4)² − x² = 8x + 16 m².
8. Master Formula Table
| Identity | Result |
|---|---|
| a(b + c) | ab + ac |
| (a + b)(c + d) | ac + ad + bc + bd |
| (a + b)² | a² + 2ab + b² |
| (a − b)² | a² − 2ab + b² |
| (a + b)(a − b) | a² − b² |
| (x + a)(x + b) | x² + (a + b)x + ab |
| a² + b² | (a + b)² − 2ab |
| (a + b)² − (a − b)² | 4ab |
9. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Distributive property | Multiply each term | Expand expressions |
| Binomial × binomial | Four products | Area models |
| Identities | True for all values | Expand, simplify |
| Mental maths | Near 100, 1000 | Quick products, squares |
| Patterns | Algebra explains them | Prove results |
10. Mind Map
- a(b + c)
- Binomial products
- (a ± b)²
- a² − b²
- (x + a)(x + b)
- Area rectangles
- Square on a + b
- Mental squares
- Products near 100
11. Tips and Tricks
- Write numbers near a round number as (round ± small) and use an identity.
- The difference of two squares factorises instantly: 87² − 13² = 100 × 74 = 7400.
- Check an expansion by substituting a small value like x = 1.
- (a + b)² = a² + b² (missing 2ab).
- (a − b)² = a² − b² (wrong).
- Sign errors when multiplying negative terms.
12. Important One-liners
- An identity is true for every value of its variables.
- (a + b)² = a² + 2ab + b².
- (a − b)² = a² − 2ab + b².
- (a + b)(a − b) = a² − b².
- (x + a)(x + b) = x² + (a + b)x + ab.
- The product of numbers 2 apart is one less than the square of the middle number.
- Difference of squares of consecutive numbers equals their sum.
13. Exam-pattern Questions with Detailed Answers
(x + 3)² = ?
- x² + 9
- x² + 6x + 9
- x² + 3x + 9
- x² + 6x + 6
Show answer
a² + 2ab + b² with a = x, b = 3.
Answer: (B)
Find 101 × 99 mentally.
Show answer
(100 + 1)(100 − 1) = 10000 − 1 = 9999.
Answer: 9999
Expand (y + 4)(y + 6).
Show answer
y² + 10y + 24.
Answer: y² + 10y + 24
Find 997² using an identity.
Show answer
(1000 − 3)² = 10,00,000 − 6000 + 9 = 9,94,009.
Answer: 9,94,009
Evaluate 78² − 22².
Show answer
(78 + 22)(78 − 22) = 100 × 56 = 5600.
Answer: 5600
Expand (3m − 2n)².
Show answer
9m² − 12mn + 4n².
Answer: 9m² − 12mn + 4n²
If x − y = 5 and xy = 6, find x² + y².
Show answer
x² + y² = (x − y)² + 2xy = 25 + 12 = 37.
Answer: 37
Which is equal to (a + b)² − (a − b)²?
- 2ab
- 4ab
- 2a² + 2b²
- 0
Show answer
(a² + 2ab + b²) − (a² − 2ab + b²) = 4ab.
Answer: (B) 4ab
Show that (n + 1)² − n² is always odd.
Show answer
(n + 1)² − n² = 2n + 1, which is odd for every whole number n.
Answer: Equals 2n + 1
Find 1.02 × 0.98 using an identity.
Show answer
(1 + 0.02)(1 − 0.02) = 1 − 0.0004 = 0.9996.
Answer: 0.9996
If a + 1/a = 4, find a² + 1/a².
Show answer
(a + 1/a)² = a² + 2 + 1/a² = 16, so a² + 1/a² = 14.
Answer: 14
The length of a rectangle is (2x + 5) and breadth (x − 1). Find its area, and the area when x = 6.
Show answer
(2x + 5)(x − 1) = 2x² − 2x + 5x − 5 = 2x² + 3x − 5. For x = 6: 72 + 18 − 5 = 85 (check: 17 × 5 = 85 ✔).
Answer: 2x² + 3x − 5; 85 square units
14. Quick Revision Summary
- Distributive property: multiply every term in one bracket by every term in the other.
- (a ± b)² = a² ± 2ab + b²; (a + b)(a − b) = a² − b²; (x + a)(x + b) = x² + (a + b)x + ab.
- Area models show why identities are true.
- Use identities for fast squares and products near round numbers.