(a - b)² Identity
The identity (a − b)² = a² − 2ab + b² gives the expansion of the square of the difference of two terms. It is the second standard algebraic identity taught in Class 8 NCERT.
This identity is closely related to the (a + b)² identity. The only difference is the sign of the middle term: in (a + b)² it is +2ab, while in (a − b)² it is −2ab.
Mastering this identity is essential for simplification, factorisation, and mental calculation of squares of numbers just below a round figure (like 98², 47², 995²).
This identity is used throughout Class 9 and 10 in polynomial factorisation, quadratic equations, and the completing-the-square method.
What is (a - b)² Identity?
Definition: The (a − b)² identity is an algebraic identity that states:
(a − b)² = a² − 2ab + b²
This identity is true for all values of a and b. It is not an equation (true for specific values) but an identity (universally true).
Key terms:
- a and b are any real numbers or algebraic expressions.
- a² is the square of the first term.
- b² is the square of the second term.
- −2ab is negative twice the product of the two terms.
- (a − b)² means (a − b) × (a − b).
Important:
- WRONG: (a − b)² = a² − b² (this confuses the identity with the difference of squares).
- RIGHT: (a − b)² = a² − 2ab + b² (note b² is POSITIVE).
(a - b)² Identity Formula
The (a − b)² Identity:
(a − b)² = a² − 2ab + b²
Where:
- a = first term
- b = second term
Useful rearrangements:
- a² + b² = (a − b)² + 2ab
- 2ab = a² + b² − (a − b)²
- (a − b) = √(a² − 2ab + b²) when a ≥ b
Relation with (a + b)²:
- (a + b)² + (a − b)² = 2(a² + b²)
- (a + b)² − (a − b)² = 4ab
Common mistakes to avoid:
- (a − b)² ≠ a² − b² (missing the −2ab term and wrong sign on b²)
- The last term b² is always positive in (a − b)², not negative.
- (a − b)² is always ≥ 0 (a square is never negative).
Derivation and Proof
Algebraic Proof:
- (a − b)² = (a − b)(a − b)
- Using the distributive property:
- = a(a − b) − b(a − b)
- = a² − ab − ba + b²
- = a² − ab − ab + b²
- = a² − 2ab + b²
Alternative derivation (using Identity I):
- Replace b with (−b) in (a + b)² = a² + 2ab + b².
- (a + (−b))² = a² + 2a(−b) + (−b)²
- = a² − 2ab + b²
- This gives the same result.
- Start with a square of side a. Its area = a².
- Remove a strip of width b from the right side: area removed = a × b = ab.
- Remove a strip of width b from the bottom: area removed = a × b = ab.
- But the corner square of side b (area = b²) was removed twice — add it back once.
- Remaining area = a² − ab − ab + b² = a² − 2ab + b².
- The remaining shape is a square of side (a − b), so its area = (a − b)².
- Therefore: (a − b)² = a² − 2ab + b².
Numerical Verification:
Let a = 7, b = 3:
- LHS = (7 − 3)² = 4² = 16
- RHS = 7² − 2(7)(3) + 3² = 49 − 42 + 9 = 16
- LHS = RHS. Verified.
Types and Properties
The (a − b)² identity is used in several types of problems:
1. Direct Expansion:
- Expand (x − 4)² = x² − 8x + 16
- Here a = x, b = 4
2. Mental Calculation of Squares:
- Find 97² = (100 − 3)² = 10000 − 600 + 9 = 9409
- Useful for numbers just below a round figure.
3. Factorisation:
- Factorise x² − 14x + 49 = (x − 7)²
- Recognise the pattern: first and last terms are perfect squares, middle term = −2ab.
4. Finding a² + b² when (a − b) and ab are known:
- Use a² + b² = (a − b)² + 2ab
5. Simplification:
- Simplify (5x − 3y)² = 25x² − 30xy + 9y²
6. Combined with other identities:
- (a + b)² − (a − b)² = 4ab
- (a + b)² + (a − b)² = 2(a² + b²)
Solved Examples
Example 1: Example 1: Direct expansion
Problem: Expand (x − 6)².
Solution:
Using (a − b)² = a² − 2ab + b²:
- Here a = x, b = 6
- (x − 6)² = x² − 2(x)(6) + 6²
- = x² − 12x + 36
Answer: (x − 6)² = x² − 12x + 36
Example 2: Example 2: Evaluating 98²
Problem: Find the value of 98² using the identity.
Solution:
- 98 = 100 − 2, so a = 100, b = 2
- 98² = (100 − 2)²
- = 100² − 2(100)(2) + 2²
- = 10000 − 400 + 4
- = 9604
Answer: 98² = 9604
Example 3: Example 3: Expansion with algebraic terms
Problem: Expand (4a − 5b)².
Solution:
- Here a = 4a, b = 5b
- (4a − 5b)² = (4a)² − 2(4a)(5b) + (5b)²
- = 16a² − 40ab + 25b²
Answer: (4a − 5b)² = 16a² − 40ab + 25b²
Example 4: Example 4: Factorisation
Problem: Factorise 9x² − 30x + 25.
Solution:
- 9x² = (3x)² — so first term = 3x
- 25 = (5)² — so second term = 5
- Middle term check: 2(3x)(5) = 30x ✓ (and sign is negative)
- This matches a² − 2ab + b² = (a − b)²
Answer: 9x² − 30x + 25 = (3x − 5)²
Example 5: Example 5: Finding a² + b²
Problem: If a − b = 5 and ab = 14, find a² + b².
Solution:
Using: a² + b² = (a − b)² + 2ab
- a² + b² = 5² + 2(14)
- = 25 + 28
- = 53
Answer: a² + b² = 53
Example 6: Example 6: Mental calculation of 47²
Problem: Find 47² without direct multiplication.
Solution:
- 47 = 50 − 3
- 47² = (50 − 3)²
- = 50² − 2(50)(3) + 3²
- = 2500 − 300 + 9
- = 2209
Answer: 47² = 2209
Example 7: Example 7: Expansion with fractions
Problem: Expand (x − 1/3)².
Solution:
- a = x, b = 1/3
- (x − 1/3)² = x² − 2(x)(1/3) + (1/3)²
- = x² − 2x/3 + 1/9
Answer: (x − 1/3)² = x² − 2x/3 + 1/9
Example 8: Example 8: Evaluating 995²
Problem: Find the value of 995² using the identity.
Solution:
- 995 = 1000 − 5
- 995² = (1000 − 5)²
- = 1000² − 2(1000)(5) + 5²
- = 1000000 − 10000 + 25
- = 990025
Answer: 995² = 990025
Example 9: Example 9: Simplifying combined expressions
Problem: Find (a + b)² − (a − b)² if a = 5, b = 3.
Solution:
Method 1 (Using identity):
- (a + b)² − (a − b)² = 4ab
- = 4 × 5 × 3
- = 60
Method 2 (Direct):
- (5 + 3)² − (5 − 3)² = 8² − 2² = 64 − 4 = 60
Answer: (a + b)² − (a − b)² = 60
Example 10: Example 10: Finding (a − b) from a² + b² and ab
Problem: If a² + b² = 85 and ab = 18, find (a − b)².
Solution:
Using: (a − b)² = a² − 2ab + b² = (a² + b²) − 2ab
- (a − b)² = 85 − 2(18)
- = 85 − 36
- = 49
Therefore, a − b = √49 = 7 (taking positive value).
Answer: (a − b)² = 49, so a − b = 7.
Real-World Applications
Mental Mathematics: This identity helps calculate squares of numbers just below a round figure. For example, 99² = (100 − 1)² = 10000 − 200 + 1 = 9801. Similarly, 48² = (50 − 2)² = 2500 − 200 + 4 = 2304.
Factorisation: Expressions of the form a² − 2ab + b² can be recognised as perfect square trinomials and factorised as (a − b)². This skill is essential in Class 9 and 10.
Completing the Square: This identity is used in the "completing the square" method for solving quadratic equations: x² − 6x + 5 = 0 → (x − 3)² − 4 = 0.
Finding Sum of Squares: When a − b and ab are given, use a² + b² = (a − b)² + 2ab to find a² + b² without knowing individual values of a and b.
Geometry: The geometric proof of this identity connects algebraic expansion with area subtraction, helping visualise algebraic concepts.
Physics: Expressions involving (v − u)² appear in equations of motion. This identity helps simplify kinematic calculations.
Key Points to Remember
- (a − b)² = a² − 2ab + b² — the second standard identity.
- The last term b² is always positive (square of b, not −b²).
- The middle term −2ab is always negative in (a − b)².
- Common mistake: (a − b)² ≠ a² − b². The correct expansion has three terms.
- (a − b)² is always ≥ 0 (a square is never negative).
- (a − b)² = 0 only when a = b.
- To factorise a² − 2ab + b², write it as (a − b)².
- a² + b² = (a − b)² + 2ab — useful when a − b and ab are known.
- (a + b)² − (a − b)² = 4ab and (a + b)² + (a − b)² = 2(a² + b²).
- Replace b with (−b) in Identity I to get Identity II.
Practice Problems
- Expand (y − 8)².
- Find the value of 96² using the identity.
- Expand (7p − 2q)².
- Factorise: 16x² − 24x + 9.
- If a − b = 4 and ab = 21, find a² + b².
- Find (a + b)² + (a − b)² when a = 6, b = 4.
- Simplify: (3x − 5)² − (3x + 5)².
- Evaluate 999² using a suitable identity.
Frequently Asked Questions
Q1. What is the (a − b)² identity?
(a − b)² = a² − 2ab + b². It gives the expansion of the square of the difference of two terms.
Q2. Is (a − b)² the same as a² − b²?
No. (a − b)² = a² − 2ab + b² (three terms with b² positive). a² − b² = (a + b)(a − b) (this is a different identity — the difference of squares).
Q3. Why is b² positive in (a − b)²?
Because (−b)² = (−b) × (−b) = +b². The square of any number (positive or negative) is always positive.
Q4. How do you use this identity for mental maths?
Write the number as (round number − small number). For example, 97² = (100 − 3)² = 10000 − 600 + 9 = 9409.
Q5. How is (a − b)² related to (a + b)²?
Replace b with −b in (a + b)² = a² + 2ab + b² to get (a − b)² = a² − 2ab + b². Also: (a + b)² − (a − b)² = 4ab and (a + b)² + (a − b)² = 2(a² + b²).
Q6. Can (a − b)² be negative?
No. (a − b)² is a perfect square and is always ≥ 0. It equals zero only when a = b.
Q7. How do you factorise using this identity?
Check if the expression matches the pattern a² − 2ab + b². If the first and last terms are perfect squares and the middle term equals −2 × (square root of first) × (square root of last), then factorise as (a − b)².
Q8. What is a² + b² in terms of (a − b)?
a² + b² = (a − b)² + 2ab. This is useful when a − b and ab are given as known values.
Q9. Is (a − b)² = (b − a)²?
Yes. (a − b)² = (b − a)² because squaring removes the sign. (a − b) = −(b − a), so (a − b)² = (−(b − a))² = (b − a)².
Q10. Give a geometric meaning of (a − b)².
Start with a square of side a (area a²). Remove two rectangular strips of dimensions a × b (area ab each). The corner square of side b (area b²) gets removed twice, so add it back once. Remaining area = a² − 2ab + b² = (a − b)².
