Expanding double brackets - harder
Expanding double brackets is a foundational skill that allows you to transform complex expressions into a simplified quadratic form, making it much easier to solve equations and sketch graphs. This process of systematic multiplication is essential for mapping out everything from the trajectory of a ball in flight to the optimized area of a construction site. Jump to the questions
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Topic guide
What this worksheet practises
This worksheet focuses on harder double-bracket expansion, where you will encounter negative terms and larger coefficients, such as (2x − 3)(3x + 4). The fundamental method is the same, but careful attention to negative numbers is required.
Key method
As with simpler brackets, you must multiply every term in the first bracket by every term in the second bracket using the FOIL method (Firsts, Outsides, Insides, Lasts). When dealing with negatives:
- Multiplying a positive and a positive gives a positive.
- Multiplying a positive and a negative gives a negative.
- Multiplying a negative and a negative gives a positive.
Worked example
Expand and simplify: (5x − 2)(x − 7).
Step 1: Firsts. Multiply 5x by x = 5x².
Step 2: Outsides. Multiply 5x by −7 = −35x.
Step 3: Insides. Multiply −2 by x = −2x.
Step 4: Lasts. Multiply −2 by −7 = +14. (A negative times a negative is a positive).
Step 5: Write out the full expression: 5x² − 35x − 2x + 14.
Step 6: Simplify by collecting the like terms (−35x − 2x = −37x).
The final answer is 5x² − 37x + 14.
Common mistakes to avoid
The most common errors involve signs. Always double-check your multiplication when negative numbers are involved, especially the "Lasts" multiplication where two negatives produce a positive. Also, be careful when collecting like terms involving negatives.