Expanding Brackets Quizzes
Visual overview of Expanding Brackets.
Introduction
Expanding brackets means removing brackets by multiplying every term inside by the term outside. It is one of the most important algebra skills in GCSE Maths, used in simplifying expressions, solving equations, factorising, and manipulating formulas. Mastering expansion builds fluency and accuracy in all algebra work.
Example: \(3(x+4)=3x+12\).
Core Concepts
Single Brackets
Multiply each term inside the bracket by the factor outside.
Formula: \(a(b+c)=ab+ac\)
- \(5(x+2)=5x+10\)
- \(-3(y-4)=-3y+12\)
- \(2(3x+5)=6x+10\)
Double Brackets (Binomial × Binomial)
Multiply each term in the first bracket by each term in the second (the FOIL method works too).
Formula: \((a+b)(c+d)=ac+ad+bc+bd\)
- \((x+3)(x+5)=x^2+5x+3x+15=x^2+8x+15\)
- \((2x-3)(x+4)=2x^2+8x-3x-12=2x^2+5x-12\)
Negative Signs
Be careful with brackets following a negative sign.
- \(-(x+5)=-x-5\)
- \(-3(x-2)=-3x+6\)
- \(-(2x-7)=-2x+7\)
Brackets with Coefficients or Powers
Distribute numbers, coefficients, or powers across terms correctly.
- \(3(2x^2+5x)=6x^2+15x\)
- \(-2(4x^3-x^2)=-8x^3+2x^2\)
Special Products
Square of a Binomial
\((a+b)^2=a^2+2ab+b^2\)
- \((x+5)^2=x^2+10x+25\)
Difference of Squares
\((a+b)(a-b)=a^2-b^2\)
- \((x+7)(x-7)=x^2-49\)
Worked Examples
Example 1 (Foundation): Single Bracket
Simplify \(4(x+3)\).
- \(4\times x=4x,\;4\times3=12\)
- Answer: \(4x+12\)
Example 2 (Foundation): Single Bracket with Negative
Simplify \(-2(y-5)\).
- \(-2y+10\)
Example 3 (Higher): Double Brackets
Simplify \((x+2)(x+3)\).
- \(x^2+5x+6\)
Example 4 (Higher): Double Brackets with Negative
Simplify \((x-4)(x+6)\).
- \(x^2+2x-24\)
Example 5 (Higher): Square of a Binomial
Simplify \((x+7)^2\).
- \(x^2+14x+49\)
Example 6 (Higher): Difference of Squares
Simplify \((x+5)(x-5)\).
- \(x^2-25\)
Example 7 (Higher): With Coefficients and Powers
Simplify \(3(2x^2+5x)\).
- \(6x^2+15x\)
Example 8 (Higher): Negative Outside a Bracket
Simplify \(-(4x^2-3x+5)\).
- \(-4x^2+3x-5\)
Example 9 (Real-Life): Rectangle Area
Length \(=(x+3)\), width \(=(x+5)\).
- Area \(=(x+3)(x+5)=x^2+8x+15\)
Common Mistakes
- Forgetting to multiply each term inside the bracket.
- Dropping or mishandling negative signs.
- Not combining like terms after expansion.
- Confusing \((a+b)^2\) with \(a^2+b^2\).
- Missing coefficient multiplications with powers.
Applications
- Algebraic simplification before solving equations.
- Geometry: area and perimeter of shapes with algebraic sides.
- Physics and economics: expanding formulas for motion, cost, or profit.
- Essential preparation for factorising and quadratic equations.
Strategies & Tips
- Multiply term-by-term—never skip any.
- Write out intermediate steps to track signs.
- Combine like terms at the end.
- Memorise special-product patterns.
- Practise FOIL expansions until automatic.
Summary / Call-to-Action
Expanding brackets converts products into sums, unlocking many algebraic methods. By mastering single and double brackets, handling negatives and coefficients, and using special-product shortcuts, students gain confidence and precision in algebra.
- Practise one- and two-bracket expansions daily.
- Check each term carefully for sign and coefficient.
- Apply special products like \((a+b)^2\) and \((a+b)(a-b)\) from memory.
- Move on to factorisation and quadratics with confidence.