(a² - b²) Identity
The difference of squares identity states that a² - b² = (a + b)(a - b). This is one of the four standard algebraic identities in Class 8 and is also called Identity III.
This identity tells us that the difference of the squares of two numbers can always be written as the product of their sum and their difference. It works for all values of a and b — integers, fractions, decimals, or algebraic expressions.
The a² - b² identity is widely used for factorisation (breaking down expressions into factors), simplification (reducing complex expressions), and quick mental arithmetic (computing products like 53 x 47 without long multiplication). It is one of the most frequently used identities in all of algebra.
What is (a² - b²) Identity?
Definition: The difference of squares identity states:
a² - b² = (a + b)(a - b)
This can be read in two ways:
- Left to right (Factorisation): a² - b² can be factored as (a + b)(a - b).
- Right to left (Expansion): (a + b)(a - b) expands to a² - b².
Important conditions:
- This identity holds for all values of a and b.
- Both terms must be perfect squares for direct factorisation (e.g., 9x² - 25 = (3x)² - 5²).
- The sign between the two squares must be minus. The identity does NOT apply to a² + b² (sum of squares cannot be factored using real numbers).
(a² - b²) Identity Formula
The Identity:
a² - b² = (a + b)(a - b)
Useful rearrangements:
- a² = b² + (a + b)(a - b) — finding a² when b², a+b, and a-b are known
- b² = a² - (a + b)(a - b)
- (a + b) = (a² - b²) / (a - b) — when a ≠ b
- (a - b) = (a² - b²) / (a + b) — when a + b ≠ 0
Connection to other identities:
- (a + b)² - (a - b)² = 4ab — this uses the difference of squares where the two squares are (a+b)² and (a-b)².
Derivation and Proof
Algebraic Derivation:
Expand (a + b)(a - b) using the distributive property:
- (a + b)(a - b) = a(a - b) + b(a - b)
- = a² - ab + ab - b²
- = a² - b²
The middle terms (-ab and +ab) cancel each other out, leaving only a² - b².
Geometric Derivation:
Consider a large square with side a (area = a²). Remove a small square with side b from one corner (area = b²). The remaining L-shaped region has area a² - b².
This L-shape can be cut and rearranged into a rectangle with:
- Length = (a + b)
- Width = (a - b)
Area of rectangle = (a + b)(a - b) = a² - b².
Numerical verification:
Let a = 7, b = 3:
- LHS: a² - b² = 49 - 9 = 40
- RHS: (a + b)(a - b) = 10 x 4 = 40
- LHS = RHS. Verified.
Types and Properties
Problems using the a² - b² identity can be classified as:
1. Factorisation:
- Rewrite expressions like x² - 49, 4m² - 9n², or 100 - y² as products of two factors.
- Identify each term as a perfect square and apply the identity.
2. Expansion:
- Expand products like (x + 5)(x - 5) directly to x² - 25.
3. Numerical calculations:
- Compute products like 103 x 97, 56 x 44, or 1001 x 999 mentally.
- Find differences of squares like 75² - 25².
4. Simplification:
- Simplify algebraic fractions where the numerator or denominator has a difference of squares.
- Cancel common factors after factorising.
5. Finding unknown values:
- Given a + b and a - b, find a² - b².
- Given a² - b² and one of the factors, find the other.
6. Multi-step factorisation:
- Expressions like x⁴ - 81 can be factored repeatedly: (x²)² - 9² = (x² + 9)(x² - 9) = (x² + 9)(x + 3)(x - 3).
Solved Examples
Example 1: Example 1: Basic factorisation
Problem: Factorise x² - 36.
Solution:
x² - 36 = x² - 6²
Using a² - b² = (a + b)(a - b) with a = x, b = 6:
= (x + 6)(x - 6)
Answer: x² - 36 = (x + 6)(x - 6).
Example 2: Example 2: Factorisation with coefficients
Problem: Factorise 25a² - 16b².
Solution:
25a² - 16b² = (5a)² - (4b)²
Using the identity with a = 5a, b = 4b:
= (5a + 4b)(5a - 4b)
Answer: 25a² - 16b² = (5a + 4b)(5a - 4b).
Example 3: Example 3: Numerical computation — product
Problem: Find the value of 103 x 97 without direct multiplication.
Solution:
Write 103 = 100 + 3 and 97 = 100 - 3.
103 x 97 = (100 + 3)(100 - 3)
Using the identity: = 100² - 3² = 10000 - 9 = 9991.
Answer: 103 x 97 = 9991.
Example 4: Example 4: Difference of two squares
Problem: Find the value of 85² - 15².
Solution:
Using a² - b² = (a + b)(a - b) with a = 85, b = 15:
= (85 + 15)(85 - 15)
= 100 x 70
= 7000
Answer: 85² - 15² = 7000.
Example 5: Example 5: Finding values from sum and difference
Problem: If a + b = 12 and a - b = 4, find a² - b².
Solution:
a² - b² = (a + b)(a - b) = 12 x 4 = 48.
Answer: a² - b² = 48.
Example 6: Example 6: Simplifying an algebraic fraction
Problem: Simplify: (x² - 9) / (x + 3).
Solution:
Factorise the numerator:
x² - 9 = x² - 3² = (x + 3)(x - 3)
So: (x² - 9) / (x + 3) = (x + 3)(x - 3) / (x + 3) = x - 3 (where x ≠ -3).
Answer: (x² - 9) / (x + 3) = x - 3 (x ≠ -3).
Example 7: Example 7: Multi-step factorisation
Problem: Factorise x⁴ - 16.
Solution:
x⁴ - 16 = (x²)² - 4²
= (x² + 4)(x² - 4)
Now x² - 4 can be factored further:
= (x² + 4)(x + 2)(x - 2)
(x² + 4 cannot be factored further using real numbers.)
Answer: x⁴ - 16 = (x² + 4)(x + 2)(x - 2).
Example 8: Example 8: Numerical — 1001 x 999
Problem: Find 1001 x 999.
Solution:
1001 x 999 = (1000 + 1)(1000 - 1)
= 1000² - 1²
= 1,000,000 - 1
= 999,999
Answer: 1001 x 999 = 999,999.
Example 9: Example 9: Factorisation with fractions
Problem: Factorise: (x²/4) - (y²/9).
Solution:
(x²/4) - (y²/9) = (x/2)² - (y/3)²
= (x/2 + y/3)(x/2 - y/3)
Answer: (x²/4) - (y²/9) = (x/2 + y/3)(x/2 - y/3).
Example 10: Example 10: Finding a missing value
Problem: If a² - b² = 56 and a - b = 4, find a + b, a, and b.
Solution:
a² - b² = (a + b)(a - b)
56 = (a + b) x 4
a + b = 56/4 = 14
Now solve: a + b = 14 and a - b = 4.
- Adding: 2a = 18, so a = 9
- Subtracting: 2b = 10, so b = 5
Verification: 9² - 5² = 81 - 25 = 56. Correct.
Answer: a + b = 14, a = 9, b = 5.
Real-World Applications
The difference of squares identity has many applications:
- Mental Arithmetic: Computing products like 52 x 48 = (50 + 2)(50 - 2) = 2500 - 4 = 2496 is much faster than long multiplication.
- Factorisation in Algebra: Many polynomial expressions can be simplified by recognising the difference of squares pattern. This is essential for solving equations, simplifying fractions, and working with rational expressions.
- Number Theory: The identity helps prove that the difference of two consecutive perfect squares is always odd: (n+1)² - n² = (n+1+n)(n+1-n) = (2n+1), which is always odd.
- Physics: In kinematics, the equation v² - u² = 2as can be factored as (v+u)(v-u) = 2as, which is useful in certain derivations.
- Trigonometry: Identities like cos²x - sin²x = cos 2x are difference-of-squares patterns.
- Higher-degree factorisation: Expressions like x⁴ - y⁴ can be factored step by step using the difference of squares repeatedly.
- Competitive Mathematics: Problems involving large differences like 9999² - 1² = (9999 + 1)(9999 - 1) = 10000 x 9998 = 99,980,000 are common in Olympiad and NTSE exams.
Key Points to Remember
- a² - b² = (a + b)(a - b) — the difference of squares identity.
- The identity works for ALL values of a and b.
- It can be used for factorisation (left to right) and expansion (right to left).
- Both terms must be perfect squares for direct application.
- The sign must be minus — a² + b² cannot be factored using this identity.
- Useful for mental calculations: (a+b)(a-b) = a² - b².
- Can be applied repeatedly for higher powers: x⁴ - y⁴ = (x² + y²)(x + y)(x - y).
- Helps simplify algebraic fractions by cancelling common factors.
- The geometric proof involves rearranging an L-shaped region into a rectangle.
- This identity is the basis for many results in number theory and trigonometry.
Practice Problems
- Factorise: 49x² - 64y².
- Find 204 x 196 using the identity.
- Evaluate: 78² - 22².
- Factorise: m⁴ - 256.
- Simplify: (4x² - 25) / (2x - 5).
- If a + b = 20 and a - b = 6, find a² - b².
- Factorise: 1 - 81a².
- Find 10.5 x 9.5 using the identity.
Frequently Asked Questions
Q1. What is the a² - b² identity?
a² - b² = (a + b)(a - b). The difference of two squares equals the product of the sum and difference of the two numbers.
Q2. Can a² + b² be factored using this identity?
No. The sum of two squares (a² + b²) cannot be factored into real linear factors. This identity applies only to the DIFFERENCE of squares.
Q3. How is this identity useful for mental calculations?
Express the numbers as (mean + deviation) and (mean - deviation). For example, 73 x 67 = (70 + 3)(70 - 3) = 4900 - 9 = 4891.
Q4. What does 'difference of squares' mean?
It means one perfect square subtracted from another. For example, 25 - 9 is a difference of squares because 25 = 5² and 9 = 3².
Q5. How do you recognise a difference of squares in an expression?
Look for two terms separated by a minus sign, where both terms are perfect squares. Examples: x² - 16, 4a² - 9b², 100 - m².
Q6. Can this identity be applied more than once?
Yes. For example, x⁴ - 81 = (x² + 9)(x² - 9) = (x² + 9)(x + 3)(x - 3). The second factor was factored again using the identity.
Q7. How is this identity derived?
Expand (a + b)(a - b) using the distributive property: a² - ab + ab - b² = a² - b². The middle terms cancel out.
Q8. What is the geometric proof?
Start with a square of side a, remove a square of side b from one corner. The remaining L-shaped area (= a² - b²) can be rearranged into a rectangle with dimensions (a+b) and (a-b).
Q9. Is (a - b)(a + b) the same as (a + b)(a - b)?
Yes. Multiplication is commutative, so the order does not matter. Both equal a² - b².
Q10. How does this identity connect to Identity I and II?
(a + b)² - (a - b)² = [a² + 2ab + b²] - [a² - 2ab + b²] = 4ab. This uses the difference of squares where the two 'squares' are (a+b)² and (a-b)².
