Expanding Single Brackets: Your Ultimate Maths Revision Guide

📚 What is Expanding Single Brackets?

Expanding single brackets, also known as removing parentheses, involves applying the distributive property of multiplication over addition or subtraction. It's a fundamental skill in algebra that simplifies expressions and allows us to solve equations. Essentially, you're multiplying each term inside the bracket by the term outside the bracket.

📜 A Brief History

The concept of expanding brackets is rooted in the development of algebraic notation. While the ancient Babylonians and Egyptians worked with algebraic concepts, the formalization of algebraic notation, including the use of parentheses and the distributive property, evolved gradually over centuries. Mathematicians like Diophantus (in ancient Greece) and later Islamic scholars contributed to the development of algebraic techniques that paved the way for modern algebra, where expanding brackets became a standard procedure.

🔑 Key Principles

• 🔢 The Distributive Property: This is the core principle. For any numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. Similarly, $a(b - c) = ab - ac$.
• ➕ Sign Rules: Remember the rules for multiplying positive and negative numbers:
  • ✅ Positive × Positive = Positive
  • ❌ Positive × Negative = Negative
  • ❌ Negative × Positive = Negative
  • ✅ Negative × Negative = Positive
• 🧮 Combining Like Terms: After expanding, simplify the expression by combining terms with the same variable and exponent (e.g., $3x + 5x = 8x$).

💡 Real-world Examples

Let's look at some examples to illustrate the process:

1. Example 1: Expand $3(x + 2)$.

   Applying the distributive property: $3(x + 2) = 3 * x + 3 * 2 = 3x + 6$
2. Example 2: Expand $-2(y - 5)$.

   Applying the distributive property: $-2(y - 5) = -2 * y + (-2) * (-5) = -2y + 10$

3. Example 3: Expand $x(2x + 3)$.

   Applying the distributive property: $x(2x + 3) = x * 2x + x * 3 = 2x^2 + 3x$

4. Example 4: Expand $4(2a - 3b + c)$.

   Applying the distributive property: $4(2a - 3b + c) = 4 * 2a - 4 * 3b + 4 * c = 8a - 12b + 4c$

✍️ Practice Quiz

Expand the following expressions:

1. $2(x + 5)$
2. $-3(y - 2)$
3. $x(x + 4)$
4. $5(2a + b)$
5. $-2(3p - q)$
6. $4(x - 2y + 3)$
7. $a(a - b + 2c)$

Answers:

1. $2x + 10$
2. $-3y + 6$
3. $x^2 + 4x$
4. $10a + 5b$
5. $-6p + 2q$
6. $4x - 8y + 12$
7. $a^2 - ab + 2ac$

🚀 Conclusion

Expanding single brackets is a fundamental algebraic skill. By understanding the distributive property and practicing regularly, you can master this concept and confidently tackle more complex algebraic problems. Keep practicing, and you'll find it becomes second nature!