KS3 Maths Expanding Double Brackets With Examples
Expanding double brackets means multiplying each term in one bracket by each term in the other. The biggest challenge is not the multiplication itself but tracking all four products and handling negative signs. A grid or a carefully organised two-line method helps students understand the distributive law and avoid missing the middle terms.
Start with a simple distributive example
Before expanding two brackets, check that the student can expand 3(x + 4) = 3x + 12. The number outside multiplies both terms inside. The same idea applies when both factors are brackets: every term in the first bracket must multiply every term in the second. Think of an area rectangle split into smaller rectangles, each representing one product. This makes the four multiplications easier to visualise.
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Expand two positive brackets
For (x + 2)(x + 3), multiply x by x to get x², x by 3 to get 3x, 2 by x to get 2x, and 2 by 3 to get 6. The expanded expression is x² + 3x + 2x + 6. Collect like terms to obtain x² + 5x + 6. Check that the coefficient of x is the sum of the two middle terms, not the product of the constant numbers.
Handle a negative term
Expand (2x − 3)(x + 4). The products are 2x², 8x, −3x and −12. Combining like terms gives 2x² + 5x − 12. The negative term −3 must multiply both terms in the second bracket; treating −3 × 4 as positive 12 would change the expression. A four-cell grid can keep each signed multiplication separate until the final collection stage.
Recognise a squared bracket
The notation (x − 3)² means (x − 3)(x − 3), not x² − 9. Expanding gives x² − 3x − 3x + 9 = x² − 6x + 9. The two middle products are important. Similarly, (x + 5)² = x² + 10x + 25. Ask students to check their result by substituting x = 0 and x = 1 into the original and expanded forms.
Difference of two squares
The expression (x + 4)(x − 4) expands to x² − 4x + 4x − 16 = x² − 16. Here the middle terms cancel. In general (a + b)(a − b) = a² − b². Recognising this structure is useful, but students should first understand why cancellation happens rather than assuming every pair of brackets loses its middle terms.
Check a completed expansion
After combining terms, substitute an easy number such as x = 2 into both expressions. For (x + 2)(x + 3), the original gives 4 × 5 = 20; the expanded x² + 5x + 6 gives 4 + 10 + 6 = 20. Agreement for one value does not prove an identity, but disagreement certainly reveals an error. Confidence with brackets supports later factorisation and quadratic equations.
Frequently asked questions
Do I multiply all four combinations?
Yes, each term in one bracket multiplies each term in the other.
Why do squared brackets have a middle term?
Because cross-products appear twice when the same two-term bracket is multiplied by itself.
Related maths learning resources
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