Difference of Two Squares

Introduction

In algebra, some expressions can be factored much more quickly when we recognise special patterns.

One of the most useful patterns is called the difference of two squares.

Consider the expression:

x² − 25

At first, this may look like an ordinary subtraction problem. However, both terms are perfect squares:

• x² = x × x

• 25 = 5 × 5

Because the two squared quantities are being subtracted, we can use a special algebraic identity:

a² − b² = (a − b)(a + b)

For example:

x² − 25 = (x − 5)(x + 5)

This identity is important because it allows us to factor expressions, simplify algebraic fractions, solve equations, and perform calculations efficiently.

Understanding Perfect Squares

Before factoring a difference of two squares, we must recognise perfect squares.

A perfect square is a number or algebraic expression obtained by multiplying a quantity by itself.

Numerical Perfect Squares

Algebraic Perfect Squares

• x² = x × x

• 4x² = (2x)²

• 9y² = (3y)²

• 16a⁴ = (4a²)²

• 25m⁶ = (5m³)²

Worked Example

Determine whether 49x² is a perfect square.

49 = 7²

x² = (x)²

Therefore:

49x² = (7x)²

49x² is a perfect square.

Important Observation

A term can be a perfect square even when it contains variables and coefficients.

Recognising the Difference of Two Squares

A difference of two squares has three important features:

• There are two terms.

• The terms are separated by subtraction.

• Both terms are perfect squares.

Examples

x² − 16

Both x² and 16 are perfect squares.

9x² − 25

Both 9x² and 25 are perfect squares.

4a² − 49b²

Both 4a² and 49b² are perfect squares.

Non-Examples

x² + 16

This is a sum of squares, not a difference.

x² − 8

The number 8 is not a perfect square of an integer, so this is not a difference of two integer perfect squares. It can still be factored over the real numbers using √8.

x² − 4x + 4

This contains three terms. It is a perfect-square trinomial rather than a two-term difference of squares.

Important Observation

The subtraction sign is essential to recognising the difference-of-squares pattern.

The Difference of Two Squares Identity

The identity is:

a² − b² = (a − b)(a + b)

This tells us that subtracting two squares is equivalent to multiplying the difference and sum of their bases.

Understanding the Geometry

Imagine a large square with side length a.

Its area is:

a²

Now imagine removing a smaller square with side length b.

The removed area is:

b²

The remaining area is:

a² − b²

For a ≥ b ≥ 0, the remaining pieces can be rearranged to form a rectangle with dimensions:

• Length = a + b

• Width = a − b

The area of this rectangle is:

(a + b)(a − b)

Therefore, the two expressions represent the same area.

Important Observation

The identity is not simply a rule to memorise. It describes a relationship that can be understood geometrically.

Deriving the Identity Algebraically

We can verify the identity using the distributive property.

Start with:

(a − b)(a + b)

Expand:

= a(a + b) − b(a + b)

= a² + ab − ab − b²

The middle terms cancel:

+ab − ab = 0

Therefore:

(a − b)(a + b) = a² − b²

This proves the identity.

Important Observation

The middle terms cancel because the two factors contain opposite signs.

Factoring Simple Expressions

Worked Example: x² − 9

Identify the squares:

x² = (x)²

9 = 3²

Apply the identity:

x² − 9 = (x − 3)(x + 3)

Final Answer: (x − 3)(x + 3)

Worked Example: x² − 64

64 = 8²

Therefore:

x² − 64 = (x − 8)(x + 8)

Final Answer: (x − 8)(x + 8)

Worked Example: y² − 100

100 = 10²

Therefore:

y² − 100 = (y − 10)(y + 10)

Final Answer: (y − 10)(y + 10)

Factoring Expressions with Coefficients

Sometimes both squared terms contain numerical coefficients.

Worked Example: 4x² − 25

Identify the square roots of the terms:

4x² = (2x)²

25 = 5²

Apply the identity:

4x² − 25 = (2x − 5)(2x + 5)

Final Answer: (2x − 5)(2x + 5)

Worked Example: 9a² − 16

9a² = (3a)²

16 = 4²

Therefore:

9a² − 16 = (3a − 4)(3a + 4)

Final Answer: (3a − 4)(3a + 4)

Worked Example: 49m² − 81

49m² = (7m)²

81 = 9²

Therefore:

49m² − 81 = (7m − 9)(7m + 9)

Final Answer: (7m − 9)(7m + 9)

Important Observation

When factoring expressions with coefficients, take the square root of the entire squared term.

Factoring Expressions with Two Variables

The identity also works when both terms contain variables.

Worked Example: x² − y²

x² = (x)²

y² = ²

Therefore:

x² − y² = (x − y)(x + y)

Worked Example: 16a² − 9b²

16a² = (4a)²

9b² = (3b)²

Therefore:

16a² − 9b² = (4a − 3b)(4a + 3b)

Worked Example: 25p⁴ − 36q²

25p⁴ = (5p²)²

36q² = (6q)²

Therefore:

25p⁴ − 36q² = (5p² − 6q)(5p² + 6q)

Important Observation

The identity applies to any expressions that can be written as squares, not only individual variables.

Verifying Factorisation

Factoring should always be checked by multiplying the factors.

Worked Example

Verify:

x² − 36 = (x − 6)(x + 6)

Expand the right-hand side:

(x − 6)(x + 6)

= x² + 6x − 6x − 36

= x² − 36

The expanded expression matches the original.

Therefore, the factorisation is correct.

Another Example

Verify:

9x² − 4 = (3x − 2)(3x + 2)

Expand:

= 9x² + 6x − 6x − 4

= 9x² − 4

The factorisation is correct.

Important Observation

Expansion and factorisation are inverse processes.

Checking Factorisation by Substitution

Another useful checking method is to substitute a numerical value.

Example

Check:

x² − 16 = (x − 4)(x + 4)

Choose x = 6.

Original expression:

6² − 16 = 36 − 16 = 20

Factored expression:

(6 − 4)(6 + 4)

= 2 × 10

= 20

Both sides give the same result.

This supports the factorisation.

However, testing one value does not prove that two expressions are identical for every value. Expanding the factors provides an algebraic proof.

Factoring Completely

Sometimes an expression has a common factor that must be removed first.

Worked Example: 2x² − 50

Both terms have a common factor of 2.

Factor out 2:

2(x² − 25)

Now apply the difference-of-squares identity:

x² − 25 = (x − 5)(x + 5)

Therefore:

2x² − 50 = 2(x − 5)(x + 5)

Worked Example: 3a² − 75b²

Factor out 3:

3(a² − 25b²)

Apply the identity:

3(a − 5b)(a + 5b)

Final Answer: 3(a − 5b)(a + 5b)

Important Observation

Always check for a greatest common factor before applying special factorisation identities.

Repeated Difference of Squares

Some expressions can be factored more than once.

Worked Example: x⁴ − 16

Recognise:

x⁴ = (x²)²

16 = 4²

Therefore:

x⁴ − 16 = (x² − 4)(x² + 4)

The first factor is another difference of squares:

x² − 4 = (x − 2)(x + 2)

Therefore:

x⁴ − 16 = (x − 2)(x + 2)(x² + 4)

This is completely factored over the real numbers using polynomial factors.

Worked Example: x⁴ − 81

x⁴ − 81 = (x² − 9)(x² + 9)

Factor x² − 9:

x⁴ − 81 = (x − 3)(x + 3)(x² + 9)

Important Observation

After factoring once, examine the resulting factors to see whether any can be factored further.

Difference of Squares with Binomials

The quantities being squared can themselves be algebraic expressions.

Worked Example

Factor:

(x + 3)² − 16

Identify:

a = x + 3

b = 4

Apply the identity:

[(x + 3) − 4][(x + 3) + 4]

Simplify:

(x − 1)(x + 7)

Final Answer: (x − 1)(x + 7)

Worked Example

Factor:

(2x − 1)² − 9

Identify:

a = 2x − 1

b = 3

Apply the identity:

[(2x − 1) − 3][(2x − 1) + 3]

Simplify:

(2x − 4)(2x + 2)

Factor out common factors:

4(x − 2)(x + 1)

Important Observation

The identity works even when a and b represent entire algebraic expressions.

Simplifying Algebraic Fractions

The difference-of-squares identity is particularly useful when simplifying rational expressions.

Worked Example

Simplify:

(x² − 25)/(x − 5)

Factor the numerator:

(x − 5)(x + 5)/(x − 5)

Cancel the common factor:

x + 5

However, the original denominator cannot equal zero.

Therefore:

Final Answer: x + 5, where x ≠ 5

Important Observation

Cancelling a common factor does not remove restrictions from the original denominator.

Simplifying More Complex Fractions

Worked Example

Simplify:

(4x² − 9)/(2x − 3)

Factor the numerator:

(2x − 3)(2x + 3)/(2x − 3)

Cancel the common factor:

2x + 3

Restriction:

2x − 3 ≠ 0

x ≠ 3/2

Final Answer

2x + 3, where x ≠ 3/2

Worked Example

Simplify:

(x² − 49)/(x² + 14x + 49)

Factor the numerator:

(x − 7)(x + 7)

Factor the denominator:

(x + 7)²

Therefore:

[(x − 7)(x + 7)]/[(x + 7)(x + 7)]

Cancel one common factor:

(x − 7)/(x + 7), where x ≠ −7

Important Observation

Only common factors may be cancelled. Individual terms connected by addition or subtraction cannot simply be cancelled.

Solving Equations Using Difference of Squares

The identity can help solve quadratic equations.

Worked Example: x² − 25 = 0

Factor:

(x − 5)(x + 5) = 0

Apply the zero-product property.

If two factors multiply to zero, at least one factor must equal zero.

Therefore:

x − 5 = 0

or

x + 5 = 0

Solutions:

x = 5 or x = −5

Verification

For x = 5:

5² − 25 = 0

For x = −5:

(−5)² − 25 = 0

Both solutions are correct.

Solving Equations with Coefficients

Worked Example

Solve:

9x² − 16 = 0

Factor:

(3x − 4)(3x + 4) = 0

Set each factor equal to zero:

3x − 4 = 0

3x = 4

x = 4/3

Or:

3x + 4 = 0

3x = −4

x = −4/3

Final Answer

x = ±4/3

Important Observation

A difference-of-squares equation set equal to zero often produces two solutions that are opposites.

Equations Requiring Rearrangement

Sometimes the equation must be rearranged before applying the identity.

Worked Example

Solve:

x² = 64

Subtract 64 from both sides:

x² − 64 = 0

Factor:

(x − 8)(x + 8) = 0

Therefore:

x = 8 or x = −8

Worked Example

Solve:

4x² = 49

Rearrange:

4x² − 49 = 0

Factor:

(2x − 7)(2x + 7) = 0

Therefore:

x = ±7/2

Applications in Geometry

The difference of two squares has a direct geometric interpretation involving areas.

Worked Example: Square Garden

A square garden has side length 12 m.

A smaller square flower bed with side length 5 m is removed from the centre.

Calculate the remaining area.

Method

Area of large square:

12² = 144 m²

Area of smaller square:

5² = 25 m²

Remaining area:

144 − 25 = 119 m²

Using the identity:

12² − 5² = (12 − 5)(12 + 5)

= 7 × 17

Remaining area = 119 m²

Important Observation

The identity can make calculations involving differences in square areas more efficient.

Applications in Mental Arithmetic

The difference-of-squares identity can simplify numerical calculations.

Worked Example: 51 × 49

Notice:

51 = 50 + 1

49 = 50 − 1

Therefore:

51 × 49 = (50 + 1)(50 − 1)

= 50² − 1²

= 2500 − 1

51 × 49 = 2499

Worked Example: 103 × 97

103 = 100 + 3

97 = 100 − 3

Therefore:

103 × 97 = 100² − 3²

= 10000 − 9

103 × 97 = 9991

Important Observation

The identity is especially useful when multiplying two numbers equally spaced above and below a convenient number.

Applications in Physics

Difference-of-squares expressions also appear in scientific equations.

Example: Kinematics

One equation of motion is:

v² = u² + 2as

where:

• v = final velocity

• u = initial velocity

• a = acceleration

• s = displacement

Rearranging:

v² − u² = 2as

Factor the left side:

(v − u)(v + u) = 2as

This shows how the difference of squared velocities can be represented as a product.

Important Observation

Algebraic identities are useful beyond mathematics because they help rearrange and simplify scientific equations.

Recognising the Pattern in More Advanced Expressions

Example: 64x⁶ − 121y⁴

Identify:

64x⁶ = (8x³)²

121y⁴ = (11y²)²

Therefore:

64x⁶ − 121y⁴ = (8x³ − 11y²)(8x³ + 11y²)

Example: 100a⁸ − b¹⁰

100a⁸ = (10a⁴)²

b¹⁰ = (b⁵)²

Therefore:

100a⁸ − b¹⁰ = (10a⁴ − b⁵)(10a⁴ + b⁵)

Important Observation

Higher powers can still form perfect squares when their exponents are even.

Common Mistakes

Confusing difference and sum of squares

x² − 9 = (x − 3)(x + 3)

But x² + 9 cannot be factored using the same identity over the real numbers.

Forgetting opposite signs

Incorrect:

(x − 5)(x − 5)

Correct:

(x − 5)(x + 5)

Forgetting to square coefficients

For 9x² − 16, the correct factors are:

(3x − 4)(3x + 4)

Failing to factor out a common factor

For 2x² − 18:

2(x² − 9) = 2(x − 3)(x + 3)

Stopping too early

For x⁴ − 16, the expression x² − 4 can be factored further.

Cancelling terms instead of factors

In algebraic fractions, cancellation is valid only for common multiplicative factors.

Forgetting denominator restrictions

When simplifying fractions, retain values excluded by the original denominator.

Forgetting negative solutions

For x² = 25:

x = 5 or x = −5

Not checking the answer

Expand the factors to verify the original expression is recovered.

Key Terms

• Perfect Square: A number or expression obtained by multiplying a quantity by itself.

• Difference of Two Squares: An expression involving the subtraction of two squared quantities.

• Identity: An equation that is true for all permitted values of its variables.

• Factorisation: Rewriting an expression as a product of factors.

• Factor: A quantity multiplied by another quantity to form a product.

• Binomial: An algebraic expression containing two terms.

• Conjugate Binomials: Binomials with identical terms but opposite signs between them.

• Expansion: Multiplying factors to produce an equivalent expression.

• Distributive Property: Rule used to multiply expressions across addition or subtraction.

• Greatest Common Factor: Largest factor common to all terms of an expression.

• Zero-Product Property: If a product equals zero, at least one factor must equal zero.

• Quadratic Equation: Equation involving a variable raised to the second power as its highest power.

• Rational Expression: Fraction containing algebraic expressions.

• Excluded Value: A value that makes an original denominator equal to zero.

• Simplification: Rewriting an expression in an equivalent, often more convenient form.

• Verification: Checking that a mathematical result is correct.

• Square Root: A value that produces a given quantity when squared.

Key Takeaways

• The difference-of-squares identity is a² − b² = (a − b)(a + b).

• Both terms must be squares, and they must be separated by subtraction.

• The two factors contain the same terms with opposite signs.

• Perfect squares can include coefficients, variables, and higher even powers.

• Factorisation can be verified by expansion.

• Always check for common factors before applying the identity.

• Some expressions require repeated factorisation.

• The identity can simplify algebraic fractions.

• Denominator restrictions must be preserved when simplifying rational expressions.

• Difference-of-squares factorisation helps solve quadratic equations.

• The identity has geometric applications involving areas.

• It can simplify mental arithmetic.

• Recognising algebraic patterns makes mathematical problem-solving more efficient.

Check Your Understanding

Recognising the Pattern

1. What is a perfect square?

2. Write the difference-of-squares identity.

3. Is x² − 49 a difference of squares?

4. Is x² + 25 a difference of squares?

5. Explain why 16a² − 81b² is a difference of squares.

Basic Factorisation

6. Factor x² − 16.

7. Factor y² − 64.

8. Factor 4x² − 9.

9. Factor 25a² − 36.

10. Factor 49m² − 100n².

Advanced Factorisation

11. Factor completely: 2x² − 32.

12. Factor completely: x⁴ − 81.

13. Factor: 16a⁴ − 25b².

14. Factor: (x + 2)² − 9.

15. Verify by expansion that (x − 7)(x + 7) = x² − 49.

Solving and Simplifying

16. Solve x² − 36 = 0.

17. Solve 4x² − 25 = 0.

18. Simplify (x² − 9)/(x − 3), stating any restrictions.

19. Calculate 52 × 48 using the difference-of-squares identity.

20. Explain why x² + 16 cannot be factored using the difference-of-squares identity over the real numbers.

Answer Key

Extended Challenge

1. Factor completely: 3x² − 147.

2. Factor completely: x⁴ − 625.

3. Factor: 81a⁶ − 64b⁴.

4. Factor completely: (2x + 1)² − 25.

5. Solve: 16x² − 81 = 0.

6. Solve: (x − 2)² − 49 = 0.

7. Simplify (9x² − 16)/(3x − 4), stating restrictions.

8. Simplify (x² − 64)/(x² + 16x + 64), stating restrictions.

9. Calculate 1004 × 996 using the difference-of-squares identity.

10. Explain why x⁴ − 16 can be factored into three real polynomial factors, while x⁴ + 16 cannot be factored directly using the same identity.

Extended Challenge Answers

Final Investigation: Exploring Square Areas

A student is investigating the relationship between two square gardens.

The larger square has side length x metres.

The smaller square has side length 4 metres.

The smaller square is removed from the larger square.

Assume x is greater than 4.

Part A: Building the Model

1. Write an expression for the area of the larger square.

2. Write an expression for the area of the smaller square.

3. Write an expression for the remaining area.

4. Factor the remaining-area expression.

5. Explain why the expression is a difference of squares.

Part B: Creating a Table

6. Calculate the remaining area when x = 5, 6, 7, 8, 9, and 10.

7. Organise the results in a table.

8. Calculate the difference in remaining area between consecutive values.

9. Determine whether the relationship between x and remaining area is linear.

10. Explain your conclusion.

Part C: Comparing Two Expressions

11. Evaluate x² − 16 when x = 7.

12. Evaluate (x − 4)(x + 4) when x = 7.

13. Compare the results.

14. Repeat for x = 10.

15. Explain why the two expressions always produce the same result.

Part D: Solving a Practical Problem

16. The remaining area is 65 m². Write an equation.

17. Solve the equation using factorisation.

18. Determine the side length of the larger square.

19. Explain why a negative solution would not be meaningful.

20. Calculate the total area of the larger square.

Investigation Answer Check

Area Model

Area of larger square:

A₁ = x²

Area of smaller square:

A₂ = 16

Remaining area:

A = x² − 16

Factor:

A = (x − 4)(x + 4)

Completed Table

The area increases by 11, 13, 15, 17, and 19 m² between consecutive side lengths.

The differences are not constant.

Therefore, the relationship is nonlinear.

Solving for 65 m²

x² − 16 = 65

x² − 81 = 0

(x − 9)(x + 9) = 0

x = 9 or x = −9

Since a physical side length cannot be negative:

x = 9 m

Area of larger square:

9² = 81 m²

Final Reflection

The difference-of-squares identity is a powerful algebraic tool that connects multiplication, factorisation, geometry, and equation solving.

By recognising when two perfect squares are separated by subtraction, we can rewrite expressions as products, simplify calculations, verify algebraic results, and solve practical problems.

The essential identity is a² − b² = (a − b)(a + b). Recognising when and how to apply it is an important foundation for more advanced algebra.