If you are a student of algebra, you might have come across the concept of factorising a quadratic expression.

Factorising is an essential skill in algebra, and it involves breaking down an algebraic expression into simpler terms.

One of the most common methods of factorising quadratic expressions is the technique known as the difference of two squares method.

This technique applies to expressions in the format:

$$ a^2 - b^2 $$

Once you identify that format, the expression can be factorised as:

$$ (a + b)(a - b) $$

How to Identify Difference in Two Squares

A difference of two squares is a specific type of binomial expression that can be factored quickly and easily, provided you correctly identify the format.

To qualify:

  • Both terms must be perfect squares
  • The operation between them must be subtraction

Examples of square numbers:
$$ 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, \ldots $$

Examples of square variables:

  • $x^2$
  • $x^2y^2$

Valid Examples:

  • $1 - x^2$
  • $4 - x^2y^2$
  • $100x^2 - 25y^2$

This method does not work if:

  • There is addition instead of subtraction
  • The terms are not perfect squares

Once identified, the factorised form is: $$ (a + b)(a - b) $$

Simple Checklist

To verify if the expression can be factorised:

  1. Are there two terms?
  2. Is it a subtraction?
  3. Is the first term a perfect square?
  4. Is the second term a perfect square?
  5. ✅ Apply the formula:
    $$ a^2 - b^2 = (a + b)(a - b) $$

Examples

Example 1

Factorise: $$ x^2 - 16 $$

Checklist:

  1. Two terms → ✅
  2. Subtraction → ✅
  3. $x^2$ is a square → ✅
  4. $16 = 4 \times 4$ → ✅

Answer: $$ (x + 4)(x - 4) $$

Example 2

Factorise: $$ 9y^2 - 64 $$

Checklist:

  1. Two terms → ✅
  2. Subtraction → ✅
  3. $9y^2 = (3y)^2$ → ✅
  4. $64 = 8^2$ → ✅

Answer: $$ (3y + 8)(3y - 8) $$

Example 3

Factorise: $$ 25 - x^2 $$

Checklist:

  1. Two terms → ✅
  2. Subtraction → ✅
  3. $25 = 5^2$ → ✅
  4. $x^2$ is a square → ✅

Answer: $$ (5 + x)(5 - x) $$

Final Notes

  • Not all expressions can be factored using this method
  • Always simplify first: remove common factors
  • Check for this pattern after simplifying

Conclusion

Mastering the difference of two squares technique:

  • Helps simplify expressions
  • Makes solving quadratic equations easier

Remember:

Only use this method if:

  • There are two terms
  • The terms are perfect squares
  • The operation is subtraction

Practice regularly to boost your confidence and problem-solving speed in algebra.