Introduction to Expanding Double Brackets

Hello, Year 8! Today, we’re going to learn about expanding double brackets. This is an important skill in maths that will help you simplify algebraic expressions. Let’s break it down step-by-step so it’s easy to understand!

What Are Double Brackets?

When we talk about double brackets, we mean something like this:

$$(a + b)(c + d)$$

Here, we have two sets of brackets. Each set contains two terms. Our job is to multiply everything in the first bracket by everything in the second bracket.

How to Expand Double Brackets

Step 1: Use the Distributive Property

To expand, we use something called the distributive property. This means you will take each term in the first bracket and multiply it by each term in the second bracket.

For example, let’s expand $$(x + 3)(x + 2)$$.

  1. Multiply the first term of the first bracket by both terms in the second bracket:
    • Multiply (x) by (x) → (x^2)
    • Multiply (x) by (2) → (2x)
  2. Multiply the second term of the first bracket by both terms in the second bracket:
    • Multiply (3) by (x) → (3x)
    • Multiply (3) by (2) → (6)
  3. Combine all the results together:
    • So, we have:
    $$x^2 + 2x + 3x + 6$$
  4. Combine like terms:
    • (2x + 3x = 5x)
    • Final answer:
    $$x^2 + 5x + 6$$

Key Rules

  1. Multiply each term in the first bracket by each term in the second bracket.
  2. Combine any like terms at the end.
  3. Always keep track of positive and negative signs!

Tips and Tricks

  1. Use FOIL Method: This stands for First, Outside, Inside, Last. It helps you remember the order to multiply:
    • First: Multiply the first terms.
    • Outside: Multiply the outer terms.
    • Inside: Multiply the inner terms.
    • Last: Multiply the last terms.
  2. Write it down: It can help to write each step down as you go. This way, you won’t miss anything!
  3. Check your work: After you’ve finished, go back and see if you can factor it back to the original brackets. If you can, you did it right!

Practice Questions

Easy Level

  1. Expand $$(x + 1)(x + 2)$$
  2. Expand $$(y + 3)(y + 4)$$
  3. Expand $$(a + 5)(a + 6)$$
  4. Expand $$(m + 2)(m + 3)$$
  5. Expand $$(p + 1)(p + 4)$$
  6. Expand $$(2 + x)(3 + x)$$
  7. Expand $$(x + 2)(x + 5)$$
  8. Expand $$(3 + z)(4 + z)$$
  9. Expand $$(x + 0)(x + 1)$$
  10. Expand $$(t + 2)(t + 3)$$
  11. Expand $$(x + 4)(x + 1)$$
  12. Expand $$(c + 1)(c + 5)$$
  13. Expand $$(d + 2)(d + 3)$$
  14. Expand $$(x + 3)(x + 0)$$
  15. Expand $$(a + 1)(a + 2)$$
  16. Expand $$(x + 1)(2x + 3)$$
  17. Expand $$(y + 2)(y + 5)$$
  18. Expand $$(x + 2)(x + 2)$$
  19. Expand $$(k + 1)(k + 4)$$
  20. Expand $$(j + 3)(j + 4)$$

Medium Level

  1. Expand $$(2x + 3)(x + 2)$$
  2. Expand $$(3a + 1)(a + 4)$$
  3. Expand $$(4y + 2)(y + 1)$$
  4. Expand $$(2p + 5)(p + 3)$$
  5. Expand $$(x – 1)(x + 4)$$
  6. Expand $$(y + 2)(y – 3)$$
  7. Expand $$(3x + 1)(2x + 2)$$
  8. Expand $$(m + 4)(m + 5)$$
  9. Expand $$(5 + a)(2 + a)$$
  10. Expand $$(n – 2)(n + 3)$$
  11. Expand $$(x + 2)(3x + 1)$$
  12. Expand $$(2x – 1)(x + 3)$$
  13. Expand $$(2y + 3)(y – 4)$$
  14. Expand $$(k + 2)(k – 3)$$
  15. Expand $$(x + 3)(x – 2)$$
  16. Expand $$(2 + m)(m + 5)$$
  17. Expand $$(3x – 2)(x + 1)$$
  18. Expand $$(4y – 1)(y + 2)$$
  19. Expand $$(3a + 2)(2a – 1)$$
  20. Expand $$(x + 5)(2x + 3)$$

Hard Level

  1. Expand $$(x + 3)(x^2 + 4x + 5)$$
  2. Expand $$(2x – 1)(3x + 4)$$
  3. Expand $$(4x + 2)(x^2 – x + 1)$$
  4. Expand $$(x – 2)(x^2 + x + 2)$$
  5. Expand $$(3a + 2)(2a – 5)$$
  6. Expand $$(x^2 + 1)(x + 2)$$
  7. Expand $$(2x + 3)(x^2 + 5)$$
  8. Expand $$(x – 3)(x^2 + 4x + 1)$$
  9. Expand $$(a + 2)(a^2 – 3a + 4)$$
  10. Expand $$(x + 5)(x^2 – x + 1)$$
  11. Expand $$(2x – 3)(x^2 + 2x + 1)$$
  12. Expand $$(3 + x)(4 + 2x)$$
  13. Expand $$(x + 1)(x^2 – 2x + 3)$$
  14. Expand $$(4x – 3)(2x + 5)$$
  15. Expand $$(x + 2)(2x^2 – x + 3)$$
  16. Expand $$(3x + 1)(x^2 + 2x + 4)$$
  17. Expand $$(x – 1)(x^2 + x + 1)$$
  18. Expand $$(5y + 1)(y^2 – y + 1)$$
  19. Expand $$(2 + 3x)(x^2 + 1)$$
  20. Expand $$(x – 4)(x^2 + 5x + 4)$$

Answers

Easy Level Answers

  1. $$x^2 + 3x + 2$$
  2. $$y^2 + 7y + 12$$
  3. $$a^2 + 11a + 30$$
  4. $$m^2 + 5m + 6$$
  5. $$p^2 + 5p + 4$$
  6. $$2x^2 + 5x + 3$$
  7. $$x^2 + 7x + 10$$
  8. $$3z + 4z + 12$$
  9. $$x^2 + x$$
  10. $$t^2 + 5t + 6$$
  11. $$x^2 + 5x + 4$$
  12. $$c^2 + 6c + 5$$
  13. $$d^2 + 5d + 6$$
  14. $$x^2 + 3x$$
  15. $$a^2 + 3a + 2$$
  16. $$2x^2 + 5x + 3$$
  17. $$y^2 + 7y + 10$$
  18. $$x^2 + 4x + 4$$
  19. $$k^2 + 5k + 4$$
  20. $$j^2 + 7j + 12$$

Medium Level Answers

  1. $$2x^2 + 7x + 6$$
  2. $$3a^2 + 13a + 4$$
  3. $$4y^2 + 10y + 2$$
  4. $$2p^2 + 19p + 15$$
  5. $$x^2 + 3x – 4$$
  6. $$y^2 – y – 6$$
  7. $$6x^2 + 5x + 2$$
  8. $$m^2 + 9m + 20$$
  9. $$a^2 + 7a + 10$$
  10. $$n^2 + n – 6$$
  11. $$3x^2 + 7x + 2$$
  12. $$2y^2 + 5y – 8$$
  13. $$k^2 – k – 6$$
  14. $$x^2 – 3x – 6$$
  15. $$2 + 5k$$
  16. $$3x^2 + 7x + 2$$
  17. $$6x^2 + 5x – 8$$
  18. $$4y^2 – 8y – 1$$
  19. $$6a^2 – 13a – 5$$
  20. $$2x^2 + 13x + 15$$

Hard Level Answers

  1. $$x^3 + 4x^2 + 5x + 9$$
  2. $$6x^2 + 5x – 4$$
  3. $$4x^3 + 2x^2 – 4x + 2$$
  4. $$x^3 – 2x^2 – 5x + 6$$
  5. $$6a^2 – 13a – 10$$
  6. $$x^3 + 2x^2 + 5x$$
  7. $$4x^3 + 6x^2 + 10$$
  8. $$x^3 + 4x^2 + 1$$
  9. $$a^3 + a^2 + 2a + 8$$
  10. $$x^3 + 5x^2 + 6x + 15$$
  11. $$2x^3 + 8x^2 + 7x – 15$$
  12. $$3x^3 + 10x^2 + 5x + 1$$
  13. $$x^3 – 2x^2 + 5x – 1$$
  14. $$5y^3 + 1y^2 + 5y – 4$$
  15. $$2x^3 + 3x^2 + 8$$
  16. $$3x^3 + 7x^2 + 12$$
  17. $$x^3 – 1x^2 + 1$$
  18. $$5y^3 – 4y^2 + 5y – 8$$
  19. $$2x^3 + 2x + 2$$
  20. $$x^3 + 5x^2 – 4x + 16$$

Conclusion

Now you know how to expand double brackets! Remember to practice as much as you can, and soon it will be second nature. Good luck, and happy expanding!