How to Expand Double Brackets (FOIL and Grid Method)
How to expand double brackets in IGCSE Maths: multiply every term in the first bracket by every term in the second, keep each sign with its term, then collect like terms. This page teaches both the FOIL method and the grid method, covers perfect squares like (x+5)² and the difference of two squares, and ends with auto-marked practice rooms from first expansions up to exam-level worded problems.
How to Expand Double Brackets using the FOIL Method
- Apply FOIL (First, Outer, Inner, Last) to generate four products.
- Write each product clearly, keeping the correct sign attached to each term.
- Collect like terms to reach the final simplified expression.
Most marks are lost on signs. Always carry the sign with the term. Students often get the first product correct but drop the sign on the second. Double-check every sign before collecting like terms.
Core Ideas for Expanding Double Brackets
Binomials (Two Terms)
A binomial has two terms, e.g. \((x+2)\). Multiplying two binomials such as \((x+2)(x+3)\) means expanding to remove the brackets and then simplifying.
FOIL = 4 Products
First Outer Inner Last
Multiply all four pairs and keep each sign attached to its term.
Collect Like Terms
Always write the uncollected line first, then combine like terms, e.g. \(3x+2x=5x\), for a clean final answer.
💡 Worked Examples (FOIL Method)
Example 1: \((x+3)(x+2)\)
Example 2: \((x-4)(x+5)\)
Example 3: \((4x+1)(3x-7)\)
Example 4: \((2x+5)(x-3)\)
How to Expand Double Brackets using the Grid Method
- Draw a \(2 \times 2\) grid. Label the rows with the terms from the first bracket and the columns with the terms from the second.
- Fill each cell by multiplying the row term by the column term.
- Write the four products side by side, then collect like terms.
| \(x\) | \(3\) | |
|---|---|---|
| \(x\) | \(x^2\) | \(3x\) |
| \(2\) | \(2x\) | \(6\) |
💡 Worked Examples (Grid Method)
Example 5: \((x+1)(x+4)\)
| \(x\) | \(4\) | |
|---|---|---|
| \(x\) | \(x^2\) | \(4x\) |
| \(1\) | \(x\) | \(4\) |
Example 6: \((x-7)(x-3)\)
| \(x\) | \(-3\) | |
|---|---|---|
| \(x\) | \(x^2\) | \(-3x\) |
| \(-7\) | \(-7x\) | \(21\) |
Special Cases: Perfect Squares and the Difference of Two Squares
A perfect square such as \((x+5)^2\) means the bracket multiplied by itself: \((x+5)^2 = (x+5)(x+5)\). Write the bracket out twice, then expand with FOIL or the grid as normal.
You cannot just square each term. Check with a number: when \(x=1\), \((1+5)^2 = 36\) but \(1^2 + 25 = 26\). Always write \((x+5)^2 = (x+5)(x+5)\) and expand fully; the middle term \(10x\) comes from the Outer and Inner products.
💡 Worked Examples (Special Cases)
Example 7: Perfect square \((x+5)^2\)
Example 8: Difference of two squares \((x+7)(x-7)\)
Example 9: Negative perfect square \((x-4)^2\)
Example 10: Quadratic term \((x^2+3)(x+2)\)
🔑 Key Points
- Every expansion produces exactly four products (FOIL is the memory aid).
- Carry the sign with the term every time.
- Always show the uncollected line before combining like terms.
- A perfect square means the bracket written twice: \((x+a)^2 = (x+a)(x+a)\).
- In \((x+a)(x-a)\) the middle terms cancel, leaving the difference of two squares \(x^2-a^2\).
⚠️ Common Pitfalls
- Forgetting the inner or outer products entirely.
- Dropping a negative sign when multiplying.
- Combining unlike terms or skipping the uncollected line.
- Writing \(x \times x\) as \(2x\) instead of \(x^2\).
- Squaring term by term: \((x+5)^2\) is not \(x^2+25\).
Ready for more practice? Work through the rooms below ↓
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