Factorising with Difference of Squares

Understanding the difference of two squares is fundamental in algebra for simplifying expressions and solving equations. This concept helps deconstruct complex algebraic structures into simpler parts.

Introduction to the Formula

• Formula: $a^2 - b^2 = (a + b)(a - b)$, a crucial tool for simplifying expressions.
• Benefits:
  • Efficiently converts expressions into product form, facilitating equation simplification.

Recognising Perfect Squares

Definition and Importance

• Perfect Square: A number or expression resulting from squaring an integer or polynomial. For example, $9 = 3^2$.
• Importance: Essential for algebraic simplification, factoring, and solving quadratic equations.

Identifying Perfect Squares

• Techniques:
  • Identify squares: Recognise $4x^2$ as $(2x)^2$.
  • Transform $x^2 + 4x + 4$ into $(x + 2)^2$.

Geometric Interpretation

• Visualise $a^2$ as a large square and $b^2$ as a smaller square.
  • The difference, $a^2 - b^2$, represents the remaining area after subtracting $b^2$ from $a^2$.

Factored Form Explanation

The algebraic identity for the difference of two squares is $a^2 - b^2 = (a + b)(a - b)$. This formula is effective in simplifying and solving equations.

Worked Examples

1. Simplify $9^2 - 4^2$:

   • Recognise that $9^2 = 81$ and $4^2 = 16$
   • Apply the formula: $81 - 16 = (9 + 4)(9 - 4)$
   • Result: $13 \times 5 = 65$

2. Factor $x^2 - 16$:

   • Identify: $x^2$ and $4^2$
   • Apply: $(x + 4)(x - 4)$

Applying the Pattern and Solving Equations

Example 1

Solve $x^2 - 25 = 0$:

• Recognise: $x^2 - 5^2$
• Factorise: $(x + 5)(x - 5) = 0$
• Solutions: $x = -5, 5$

Example 2

Solve $9x^2 - 49 = 0$:

• Recognise: $(3x)^2 - 7^2$
• Factor: $(3x + 7)(3x - 7) = 0$
• Solutions: $x = -\frac{7}{3}, \frac{7}{3}$

Common Errors and How to Avoid Them

• Mistakes:
  • Misapplying to sums such as $a^2 + b^2$.
  • Confusing with other identities.
• Tips:
  • Apply only to differences of squares.
  • Confirm terms are perfect squares before applying the formula.

Exercises with Solutions

• Factorise $x^2 - 36$.

  • Solution: $(x + 6)(x - 6)$

• Simplify $16x^4 - 25$.

  • Solution: $(4x^2 + 5)(4x^2 - 5)$

• Factorise $49a^2 - 36b^2$.

  • Solution: $(7a + 6b)(7a - 6b)$

Conclusion

Recognising and applying the difference of two squares is crucial for efficiently simplifying expressions and solving equations. Regular practice will enhance understanding and application.