Algebra

What you'll learn

Algebra forms a substantial portion of both Foundation and Higher tier Edexcel GCSE Mathematics papers, typically accounting for 30-35% of available marks. This guide covers simplifying expressions, solving equations and inequalities, working with sequences, graphing linear and quadratic functions, and manipulating formulae. Mastery of these skills is essential for success across all three examination papers.

Key terms and definitions

Expression — a mathematical statement containing variables, numbers and operations but no equals sign (e.g. 3x + 5)

Equation — a statement showing that two expressions are equal, containing an equals sign and solvable to find unknown values

Coefficient — the numerical factor multiplying a variable (in 5x², the coefficient of x² is 5)

Like terms — terms containing identical variables raised to the same powers, which can be combined through addition or subtraction

Linear equation — an equation where the highest power of the variable is 1, producing a straight-line graph

Quadratic equation — an equation in the form ax² + bx + c = 0 where a ≠ 0

Inequality — a statement comparing two expressions using <, >, ≤ or ≥ symbols

Substitution — replacing variables with given numerical values to evaluate an expression

Core concepts

Simplifying algebraic expressions

Collecting like terms requires identifying terms with identical variable components. Only coefficients of like terms can be added or subtracted:

• 4x + 3y - 2x + 5y = 2x + 8y
• 6a² + 2a - 3a² + 7 = 3a² + 2a + 7

Expanding brackets uses the distributive law. Each term outside the bracket multiplies every term inside:

• Single brackets: 3(2x + 5) = 6x + 15
• Two brackets: (x + 4)(x + 3) = x² + 3x + 4x + 12 = x² + 7x + 12
• Use the FOIL method (First, Outer, Inner, Last) for double brackets

Factorising reverses expansion by extracting common factors:

• Single factor: 6x + 9 = 3(2x + 3)
• Quadratic expressions: x² + 7x + 12 = (x + 3)(x + 4)
• Difference of two squares: x² - 16 = (x + 4)(x - 4)

For quadratic factorisation, find two numbers that multiply to give the constant term and add to give the coefficient of x.

Solving linear equations

The goal is isolating the variable by performing inverse operations equally to both sides. Edexcel GCSE Mathematics papers test increasingly complex arrangements:

One-step equations:

• x + 7 = 12 → x = 5
• 3x = 21 → x = 7

Multi-step equations:

1. Expand brackets if present
2. Collect variable terms on one side
3. Collect number terms on the other side
4. Divide by the coefficient

Example: 3(x + 2) = 18

• 3x + 6 = 18
• 3x = 12
• x = 4

Equations with unknowns on both sides:

5x + 3 = 2x + 15

• 5x - 2x = 15 - 3
• 3x = 12
• x = 4

Equations with fractions require finding a common denominator or multiplying through:

x/4 + 2 = 5

• x/4 = 3
• x = 12

Solving quadratic equations

Edexcel papers expect students to solve quadratics using multiple methods:

Factorisation method (when the quadratic factorises neatly):

1. Rearrange to standard form (= 0)
2. Factorise the quadratic
3. Set each factor equal to zero
4. Solve for x

Example: x² + 5x + 6 = 0

• (x + 2)(x + 3) = 0
• x + 2 = 0 or x + 3 = 0
• x = -2 or x = -3

Quadratic formula (for any quadratic equation):

For ax² + bx + c = 0:

x = [-b ± √(b² - 4ac)] / 2a

This formula is provided in the exam but students must substitute correctly and simplify accurately.

Completing the square transforms x² + bx + c into (x + p)² + q form, useful for finding turning points and solving equations.

Inequalities

Solving inequalities follows the same process as equations with one critical rule: multiplying or dividing by a negative number reverses the inequality sign.

• 3x + 5 > 17 → 3x > 12 → x > 4
• -2x ≤ 8 → x ≥ -4 (sign reversed)

Represent solutions on number lines using:

• Open circles (○) for < or >
• Closed circles (●) for ≤ or ≥
• Shaded regions showing solution sets

Inequality notation may use set notation for Higher tier:

• {x: x > 4} means "the set of all x values where x is greater than 4"

Sequences

Edexcel GCSE Mathematics tests both term-to-term and position-to-term rules.

Arithmetic sequences have a common difference between consecutive terms:

• Sequence: 3, 7, 11, 15, ...
• First term (a) = 3, common difference (d) = 4
• nth term formula: Un = a + (n - 1)d
• Un = 3 + (n - 1)4 = 4n - 1

Geometric sequences have a common ratio:

• Sequence: 2, 6, 18, 54, ...
• First term = 2, common ratio = 3
• nth term: Un = ar^(n-1) = 2 × 3^(n-1)

Quadratic sequences have a second difference that is constant:

1. Find first differences
2. Find second differences
3. If second difference = 2a, nth term begins with an²
4. Subtract this from original sequence and find linear pattern

Graphs of functions

Linear graphs take the form y = mx + c where:

• m = gradient (steepness)
• c = y-intercept (where the line crosses the y-axis)

Calculate gradient from two points (x₁, y₁) and (x₂, y₂):

m = (y₂ - y₁)/(x₂ - x₁)

Parallel lines have identical gradients. Perpendicular lines have gradients that multiply to give -1.

Quadratic graphs produce U-shaped (or inverted U) curves called parabolas:

• y = x² has vertex at origin, opens upward
• y = -x² opens downward
• y = x² + 3 shifts the graph up 3 units
• y = (x - 2)² shifts the graph right 2 units

Find coordinates by substituting x values into the equation and calculating corresponding y values. Plot at least 5 points and draw a smooth curve.

Cubic and reciprocal graphs appear on Higher tier:

• y = x³ produces an S-shaped curve
• y = 1/x produces a hyperbola with two separate branches

Changing the subject of a formula

Rearranging formulae uses the same inverse operations as solving equations:

Make r the subject of A = πr²:

1. A/π = r²
2. r = √(A/π)

Make x the subject of y = (x + 3)/5:

1. 5y = x + 3
2. x = 5y - 3

For Higher tier, formulae may require factorising when the required subject appears multiple times:

Make t the subject of v = u + at:

• at = v - u
• t = (v - u)/a

Make p the subject of 3p + 2q = p - 5:

• 3p - p = -5 - 2q
• 2p = -5 - 2q
• p = (-5 - 2q)/2

Worked examples

Example 1: Solving an equation with brackets and unknowns on both sides (4 marks)

Solve 4(2x - 3) = 3x + 7

Solution:

Expand brackets: 8x - 12 = 3x + 7 ✓

Collect x terms: 8x - 3x = 7 + 12 ✓

Simplify: 5x = 19 ✓

Divide: x = 19/5 or x = 3.8 ✓

Example 2: Finding the nth term of a sequence (3 marks)

Find the nth term of the sequence 5, 8, 11, 14, 17, ...

Solution:

First difference = 3 (sequence is arithmetic) ✓

General form: 3n + c

When n = 1: 3(1) + c = 5, so c = 2 ✓

nth term = 3n + 2 ✓

Example 3: Solving a quadratic equation by factorisation (3 marks)

Solve x² - 7x + 12 = 0

Solution:

Factorise: (x - 3)(x - 4) = 0 ✓

Set each bracket to zero: x - 3 = 0 or x - 4 = 0 ✓

Solutions: x = 3 or x = 4 ✓

Common mistakes and how to avoid them

• Mistake: Forgetting to multiply all terms when expanding brackets, e.g. writing 3(x + 4) = 3x + 4. Correction: Multiply every term inside the bracket: 3(x + 4) = 3x + 12.

• Mistake: Only performing operations on one side of an equation. Correction: Whatever you do to one side, you must do to the other to maintain equality. Show each step clearly.

• Mistake: Incorrectly combining unlike terms, such as adding 3x + 4y to get 7xy. Correction: Only terms with identical variable parts can be combined. 3x + 4y cannot be simplified further.

• Mistake: Reversing inequality signs when multiplying or dividing by positive numbers. Correction: Only reverse the sign when multiplying or dividing both sides by a negative number.

• Mistake: Using the quadratic formula incorrectly by substituting b as a positive when it's negative in the original equation. Correction: Identify a, b and c from the standard form ax² + bx + c = 0, including their signs, before substituting.

• Mistake: Stopping after finding one solution to a quadratic equation. Correction: Quadratic equations typically have two solutions. Always check both factors equal to zero.

Exam technique for Algebra

• Show all working clearly. Even if the final answer is incorrect, method marks are awarded for correct processes. Set out multi-step solutions line by line.

• Check answers make sense. Substitute solutions back into the original equation to verify. For inequalities, test a value from your solution set.

• Command words matter. "Simplify" means combine like terms and reduce to simplest form. "Solve" requires finding the value(s) of the variable. "Factorise" means write as a product of factors.

• Marks allocation guides time. A 4-mark question expects 4 distinct steps or pieces of working. Don't spend 10 minutes on a 2-mark question.

Quick revision summary

Algebra at GCSE level requires fluency in manipulating expressions, solving linear and quadratic equations, working with sequences and their nth terms, sketching and interpreting graphs, and rearranging formulae. Master factorisation and expanding brackets as these underpin many topics. Remember the quadratic formula and how to apply it. Practice showing clear, step-by-step working as method marks are available even when final answers are incorrect. Always perform the same operation to both sides of equations and inequalities, reversing inequality signs only when multiplying or dividing by negatives.