Difference of Squares - Applications
The difference of two squares identity is one of the most useful algebraic identities for mental maths and simplification. It states that a² − b² = (a + b)(a − b).
This identity allows us to quickly compute products like 53 × 47, 102 × 98, or 997 × 1003 without long multiplication. It also helps in factorising expressions and simplifying calculations in geometry and physics.
In this topic, we focus on practical applications of the a² − b² identity — mental arithmetic tricks, algebraic simplification, and real-world problem solving.
What is Difference of Squares - Applications?
Definition: The difference of two squares identity states:
a² − b² = (a + b)(a − b)
This means:
- The difference of two perfect squares can always be written as a product of two factors.
- The factors are the sum and difference of the two numbers.
- Conversely, a product of the form (a + b)(a − b) always equals a² − b².
Difference of Squares - Applications Formula
Key Formulas:
a² − b² = (a + b)(a − b)
Applications for mental maths:
- To find n²: Write n as (a + b) or (a − b) where a is a round number. Then n² = a² ± 2ab + b², or use the difference of squares.
- To multiply (a + b)(a − b): The answer is simply a² − b².
- To compute 97²: Write as (100 − 3)² = 10000 − 600 + 9 = 9409. Alternatively, 97² = 100² − 3² − 2(3)(100 − 97) ... but the direct (a−b)² method is cleaner.
Derivation and Proof
Why does a² − b² = (a + b)(a − b)?
- Start with the right side: (a + b)(a − b).
- Expand using distribution (FOIL):
(a + b)(a − b) = a·a − a·b + b·a − b·b - Simplify: = a² − ab + ab − b²
- The middle terms cancel: = a² − b²
- This proves the identity.
Geometric interpretation:
- A square of side a has area a².
- Remove a smaller square of side b from one corner. The remaining area is a² − b².
- This L-shaped region can be rearranged into a rectangle of dimensions (a + b) × (a − b).
Types and Properties
Types of applications:
- Type 1 — Quick multiplication: Products like 52 × 48 = (50 + 2)(50 − 2) = 50² − 2² = 2500 − 4 = 2496.
- Type 2 — Finding squares mentally: 99² = (100 − 1)(100 + 1) + 1 = 10000 − 1 = 9999? No — 99² = (100)² − 2(100)(1) + 1² = 9801. Use (a−b)² for squaring.
- Type 3 — Factorising expressions: 4x² − 9 = (2x)² − 3² = (2x + 3)(2x − 3).
- Type 4 — Simplifying fractions: (x² − 25)/(x + 5) = (x+5)(x−5)/(x+5) = x − 5.
- Type 5 — Number theory: The difference of squares of two consecutive numbers: n² − (n−1)² = 2n − 1 (always odd).
Solved Examples
Example 1: Example 1: Quick multiplication using identity
Problem: Find 53 × 47 without long multiplication.
Solution:
Observation: 53 = 50 + 3 and 47 = 50 − 3.
- 53 × 47 = (50 + 3)(50 − 3)
- = 50² − 3²
- = 2500 − 9
- = 2491
Answer: 53 × 47 = 2491.
Example 2: Example 2: Product of numbers equidistant from 100
Problem: Find 102 × 98.
Solution:
Observation: 102 = 100 + 2 and 98 = 100 − 2.
- 102 × 98 = (100 + 2)(100 − 2)
- = 100² − 2²
- = 10000 − 4
- = 9996
Answer: 102 × 98 = 9996.
Example 3: Example 3: Large number multiplication
Problem: Find 997 × 1003.
Solution:
Observation: 997 = 1000 − 3 and 1003 = 1000 + 3.
- 997 × 1003 = (1000 − 3)(1000 + 3)
- = 1000² − 3²
- = 1000000 − 9
- = 999991
Answer: 997 × 1003 = 999991.
Example 4: Example 4: Evaluating 78² − 22²
Problem: Evaluate 78² − 22².
Solution:
Using a² − b² = (a + b)(a − b):
- 78² − 22² = (78 + 22)(78 − 22)
- = 100 × 56
- = 5600
Answer: 78² − 22² = 5600.
Example 5: Example 5: Factorising an algebraic expression
Problem: Factorise 9x² − 16y².
Solution:
- 9x² = (3x)² and 16y² = (4y)²
- 9x² − 16y² = (3x)² − (4y)²
- = (3x + 4y)(3x − 4y)
Answer: 9x² − 16y² = (3x + 4y)(3x − 4y).
Example 6: Example 6: Simplifying a fraction
Problem: Simplify (x² − 49)/(x − 7).
Solution:
- x² − 49 = x² − 7² = (x + 7)(x − 7)
- (x² − 49)/(x − 7) = (x + 7)(x − 7)/(x − 7)
- = x + 7 (for x ≠ 7)
Answer: (x² − 49)/(x − 7) = x + 7.
Example 7: Example 7: Finding a number
Problem: The difference of squares of two numbers is 80. If the larger number is 9, find the smaller.
Solution:
Given: a² − b² = 80, a = 9.
- 81 − b² = 80
- b² = 81 − 80 = 1
- b = 1
Answer: The smaller number is 1.
Example 8: Example 8: Consecutive numbers
Problem: Prove that the difference of squares of two consecutive numbers is always odd.
Solution:
Let the consecutive numbers be n and n + 1.
- (n + 1)² − n² = (n + 1 + n)(n + 1 − n)
- = (2n + 1)(1)
- = 2n + 1
Since 2n + 1 is always odd (one more than an even number), the difference is always odd.
Answer: Proved. The difference equals 2n + 1, which is always odd.
Example 9: Example 9: Multi-step calculation
Problem: Evaluate 105² − 95².
Solution:
- 105² − 95² = (105 + 95)(105 − 95)
- = 200 × 10
- = 2000
Answer: 105² − 95² = 2000.
Example 10: Example 10: Application in geometry
Problem: A square garden has side 25 m. A path of width 5 m runs along two adjacent sides (outside). Find the area of the path using the identity.
Solution:
Given: Inner square side = 25 m, path width = 5 m.
- The outer region along two sides can be computed as difference of areas.
- Total outer area (one L-strip): 30 × 30 − 25 × 25 = (30 + 25)(30 − 25) = 55 × 5 = 275 m²
Wait — this gives the area of the border on ALL sides. For two sides only, a different approach is needed. Let us reconsider:
- Area of full outer square = 30² = 900 m²
- Area of inner square = 25² = 625 m²
- Area of border (all 4 sides) = 900 − 625 = 275 m²
- Using identity: 30² − 25² = (30+25)(30−25) = 55 × 5 = 275 m²
Answer: Area of the border = 275 m².
Real-World Applications
Real-world applications:
- Mental arithmetic: Quickly multiply numbers that are equidistant from a round number (e.g., 48 × 52, 97 × 103).
- Factorisation: Factorise algebraic expressions of the form a² − b².
- Simplification: Cancel common factors in fractions involving differences of squares.
- Geometry: Calculate areas of annular (ring-shaped) regions: π(R² − r²) = π(R+r)(R−r).
- Number patterns: Prove properties of odd and even numbers.
- Physics: Kinetic energy differences, lens equations, and other formulas involve a² − b².
Key Points to Remember
- a² − b² = (a + b)(a − b) — learn this identity thoroughly.
- Use it for quick multiplication of numbers equidistant from a round number.
- The difference of squares of consecutive integers is always odd: (n+1)² − n² = 2n + 1.
- To evaluate expressions like 78² − 22², factorise first — it is much faster than squaring both numbers.
- The identity also works for algebraic expressions: 4x² − 9 = (2x+3)(2x−3).
- The geometric interpretation: removing a square from a square gives an L-shape that can be reshaped into a rectangle.
- Ring area = π(R² − r²) = π(R + r)(R − r).
- Always check if an expression is a difference of perfect squares before attempting to factorise.
Practice Problems
- Evaluate 73 × 67 using the difference of squares identity.
- Evaluate 201 × 199.
- Find 85² − 15².
- Factorise: 25a² − 36b².
- Simplify: (4x² − 1)/(2x + 1).
- The product of two numbers is 2491 and their sum is 100. Find the numbers.
- Prove that the difference of squares of any two odd numbers is divisible by 8.
- Evaluate: 10.5² − 9.5².
Frequently Asked Questions
Q1. What is the difference of two squares identity?
a² − b² = (a + b)(a − b). The difference of two squares equals the product of their sum and their difference.
Q2. How is this useful for mental maths?
When multiplying two numbers equidistant from a round number (like 48 × 52 = 50² − 2² = 2496), you avoid long multiplication entirely.
Q3. Does this identity work for non-integers?
Yes. It works for all real numbers, including decimals and fractions. For example, (3.5)² − (1.5)² = (3.5 + 1.5)(3.5 − 1.5) = 5 × 2 = 10.
Q4. Can a sum of two squares be factorised similarly?
No. a² + b² cannot be factorised into real linear factors. Only the DIFFERENCE of two squares can be factorised as (a+b)(a−b).
Q5. What if b > a in a² − b²?
Then a² − b² is negative. The identity still holds: a² − b² = (a+b)(a−b), which will be negative since (a−b) < 0.
Q6. How is this related to factorisation?
When you see an expression like 16x² − 25, recognise it as (4x)² − 5² and factorise as (4x + 5)(4x − 5). This is a key factorisation technique.
Q7. What is the geometric meaning?
If you cut a b × b square from a corner of an a × a square, the remaining L-shaped area (a² − b²) can be rearranged into a rectangle of dimensions (a + b) by (a − b).
Q8. Why is the difference of consecutive squares always odd?
(n+1)² − n² = (n+1+n)(n+1−n) = (2n+1)(1) = 2n+1, which is always odd.
