Sheet № 12 · Foundation + Higher · AQA · Edexcel · OCR
Expanding Brackets
Expanding brackets is a core algebra skill tested at both Foundation and Higher tier in GCSE Maths. It involves multiplying out expressions to remove brackets, a process you will use when solving equations, simplifying expressions, proving algebraic identities, and working with quadratics. Foundation students need to expand single bracket
§Key definitions
Example:
Expand 3(2x + 5).
Question:
Expand and simplify 5(2x − 3) + 2(x + 4).
Answer:
2x² + 7x − 15
Q1 (Foundation):
Expand 4(3x − 2).
Q2 (Foundation):
Expand and simplify (x + 6)(x − 2).
§Formulas to memorise
Single bracket: a(b + c) = ab + ac
Double brackets: (a + b)(c + d) = ac + ad + bc + bd
Squaring a bracket: (a + b)² = a² + 2ab + b²
Difference of two squares: (a + b)(a − b) = a² − b²
3 × 2x = 6x
3 × 5 = 15
x × x = x²
x × 7 = 7x
3 × x = 3x
3 × 7 = 21
Collect like terms: x² + 7x + 3x + 21 = x² + 10x + 21
4(x + 2) = 4x + 8
Worked example
Expand and simplify 5(2x − 3) + 2(x + 4).
Working:
⚠ Common mistakes
- ✗Not multiplying every term inside the bracket. In 3(x + 4), students sometimes write 3x + 4 instead of 3x + 12. The 3 must multiply both x and 4.
- ✗Sign errors with negative terms. When expanding −2(x − 5), remember: −2 × x = −2x and −2 × (−5) = +10. A negative times a negative is a positive.
- ✗Writing (x + 3)² as x² + 9. You must expand as (x + 3)(x + 3) = x² + 6x + 9. The middle term (2ab) is always present. See simplifying expressions for more on combining like terms after expanding.
- ✗Forgetting to collect like terms. After expanding double brackets, you will usually have two like terms in the middle that need combining. Leaving them uncombined when the question says "simplify" will cost marks.
- ✗Mixing up signs in the FOIL method. Track each multiplication carefully. Writing out F, O, I, L labels beside your working can help.
✦ Exam tips
- →Use a grid or table for double brackets if FOIL confuses you. Draw a 2×2 grid with one bracket across the top and the other down the side. Multiply to fill in the four cells, then add them up. This is a reliable, visual method.
- →For "show that" or "prove" questions, you must show each expansion step. Jumping to the final answer without showing the multiplication earns zero marks even if the answer is correct.
- →Practise squaring brackets until (a + b)² = a² + 2ab + b² is automatic. This pattern appears in completing the square, the quadratic formula, and circle equations. Our formulas guide lists the identities to learn.
- →Check your expansion by substituting a value. For example, expand (x + 2)(x + 3) to get x² + 5x + 6. Substitute x = 1: (3)(4) = 12, and 1 + 5 + 6 = 12. They match, so the expansion is correct.