Simplifying Expressions Quizzes
Visual overview of Simplifying Expressions.
Introduction
Simplifying expressions means rewriting an expression in a shorter, clearer form without changing its value. It underpins solving equations, expanding and factorising, and efficient calculation—key GCSE skills.
Example: \(2x+3x=5x\). Spotting like terms makes work faster and less error-prone.
Core Concepts
Like Terms
Like terms have the same variables with the same powers (order may differ). Only like terms combine.
- \(3x+5x=8x\)
- \(2y+7y=9y\)
- Non-example: \(2x+3y\) cannot combine
Coefficients and Variables
Each term = coefficient × variable part.
- In \(4x\): coefficient \(=4\), variable \(=x\)
- In \(-3y\): coefficient \(=-3\), variable \(=y\)
Combining Like Terms
- Group like terms
- Add/subtract coefficients
- Keep the variable part the same
Examples
- \(2x+5x-3x=(2+5-3)x=4x\)
- \(3a+2b-a+5b=(3-1)a+(2+5)b=2a+7b\)
Expanding Single Brackets
Distributive property: \(a(b+c)=ab+ac\).
- \(3(x+4)=3x+12\)
- \(-2(y-5)=-2y+10\)
- \(4(2x+3y)=8x+12y\)
Expanding Double Brackets
Distribute each term in the first bracket across the second (FOIL is a memory aid):
\((a+b)(c+d)=ac+ad+bc+bd\)
- \((x+3)(x+5)=x^2+5x+3x+15=x^2+8x+15\)
- \((2x-3)(x+4)=2x^2+8x-3x-12=2x^2+5x-12\)
Factorising (Reverse of Expanding)
Take out the greatest common factor (GCF) of all terms.
- \(6x+9=3(2x+3)\)
- \(8a+12b=4(2a+3b)\)
Negative Signs
- \(-2x+5x=3x\)
- \(-(3x+4)=-3x-4\)
- \(-(-2x+5)=2x-5\)
Substitution
After simplifying, substitute values and evaluate.
- For \(2x+3y\) with \(x=4,\,y=5\): \(2(4)+3(5)=8+15=23\)
Indices (Powers) in Expressions
Use index laws before combining.
- \(x^m\cdot x^n=x^{m+n}\)
- \((x^m)^n=x^{mn}\)
- \((x^a y^b)(x^c y^d)=x^{a+c}y^{b+d}\)
- \(x^2\cdot x^3=x^{5}\)
- \((x^4y^2)(x^3y)=x^{7}y^{3}\)
Worked Examples
Example 1 (Foundation): Combine Like Terms
Simplify \(3x+5x-2x\).
- \(3+5-2=6\) → answer \(=6x\)
Example 2 (Foundation): Expand a Single Bracket
Simplify \(4(x+7)\).
- \(4x+28\)
Example 3 (Higher): Expand Double Brackets
Simplify \((x+3)(x+4)\).
- \(x^2+4x+3x+12=x^2+7x+12\)
Example 4 (Higher): Factorise
Simplify \(12x+18\).
- GCF \(=6\) → \(6(2x+3)\)
Example 5 (Higher): Mind the Minus
Simplify \(-3x+5x-(-2x)\).
- \(-3x+5x=2x\); minus a negative \(=+2x\) → \(4x\)
Example 6 (Higher): Substitution
For \(2x+3y\) with \(x=4,\,y=5\): \(23\).
Example 7 (Higher): Indices
Simplify \((x^2\cdot x^3)+2x^4\).
- \(x^5+2x^4\) (not combinable)
Example 8 (Higher): Mixed Expand & Combine
Simplify \(3(x+2)-2(x-5)\).
- \(3x+6-2x+10=x+16\)
Common Mistakes
- Combining unlike terms (e.g. \(2x+3y\rightarrow 5xy\) ❌)
- Forgetting to distribute a negative across brackets
- Misusing index laws (e.g. \(x^2+x^3=x^5\) ❌)
- Not taking the greatest common factor when factorising
- Substituting before simplifying (causes messy arithmetic)
Applications
- Solving linear and quadratic equations/inequalities
- Geometry (perimeters, areas) and probability expressions
- Cost, distance, and rate models
- Preparing expressions for substitution and graphing
Strategies & Tips
- Underline like terms with matching colours/marks.
- Add invisible 1s for clarity: \(x = 1x\), \(-x = -1x\).
- Use brackets to control negatives and long expressions.
- Check each step (especially signs) before moving on.
- Practise factorising and double-bracket expansion regularly.
Summary / Call-to-Action
Fluent simplification—combining like terms, careful expansion, correct factorisation, and index laws—makes algebra quicker and safer.
- Drill single- and double-bracket expansions.
- Practise GCF factorising daily.
- Mix negatives and indices to build accuracy.