Algebra

What you'll learn

Algebra forms a substantial component of both Foundation and Higher tier Edexcel GCSE Mathematics papers. This revision guide covers all testable algebraic content from simplifying expressions through to solving complex equations, working with sequences, manipulating formulae and interpreting graphs. Mastering these techniques is essential for accessing marks across all three exam papers.

Key terms and definitions

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

Equation — a mathematical statement showing that two expressions are equal, containing an equals sign (e.g. 3x + 5 = 17)

Coefficient — the numerical factor multiplied by a variable (in 7x², the coefficient is 7)

Linear equation — an equation where the highest power of the variable is 1 (e.g. 5x - 3 = 12)

Quadratic expression — an expression containing a term in x² as the highest power (e.g. x² + 7x + 10)

Factorising — writing an expression as a product of its factors (the reverse of expanding brackets)

Simultaneous equations — two or more equations that are true at the same time, sharing common variables

Inequality — a mathematical statement using <, >, ≤ or ≥ symbols to show one expression is larger or smaller than another

Core concepts

Simplifying and manipulating expressions

Collect like terms by adding or subtracting terms with identical variable parts. Only terms with exactly the same letter combination can be combined.

Examples of collecting like terms:

• 5x + 3x - 2x = 6x
• 4a + 3b - 2a + 5b = 2a + 8b
• 3x² + 5x - 2x² + 3x = x² + 8x

Expanding single brackets:

Multiply every term inside the bracket by the term outside:

• 4(2x + 3) = 8x + 12
• -3(5x - 2) = -15x + 6
• x(x + 4) = x² + 4x

Expanding double brackets:

Use the grid method or FOIL (First, Outer, Inner, Last):

• (x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15
• (2x - 1)(x + 4) = 2x² + 8x - x - 4 = 2x² + 7x - 4
• (x - 3)(x - 7) = x² - 7x - 3x + 21 = x² - 10x + 21

Factorising expressions:

Always look for common factors first, then consider other methods.

Single brackets (common factors):

• 6x + 9 = 3(2x + 3)
• 4x² - 6x = 2x(2x - 3)
• 15ab - 10a = 5a(3b - 2)

Quadratic expressions (two brackets):

• x² + 7x + 12 = (x + 3)(x + 4)
• x² - 5x + 6 = (x - 2)(x - 3)
• x² + 3x - 10 = (x + 5)(x - 2)

Difference of two squares:

• x² - 25 = (x + 5)(x - 5)
• 9x² - 16 = (3x + 4)(3x - 4)

Solving linear equations

The fundamental principle is to perform inverse operations to isolate the variable, maintaining balance by doing the same to both sides.

Single-step equations:

• x + 7 = 15 → x = 8
• 3x = 21 → x = 7

Multi-step equations:

1. Expand any brackets
2. Collect variable terms on one side, numbers on the other
3. Simplify to find x

Example: 3(2x - 1) = 15

• 6x - 3 = 15
• 6x = 18
• x = 3

Equations with unknowns on both sides:

• 5x + 3 = 2x + 18
• 3x + 3 = 18
• 3x = 15
• x = 5

Equations with fractions:

Multiply through by the denominators to clear fractions:

• (x + 2)/3 = 5
• x + 2 = 15
• x = 13

Solving quadratic equations

Factorising method (when factorisable):

1. Rearrange to = 0
2. Factorise
3. Set each bracket = 0
4. Solve each linear equation

Example: x² + 5x + 6 = 0

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

Quadratic formula (any quadratic):

For ax² + bx + c = 0:

x = [-b ± √(b² - 4ac)] / 2a

This formula is given on the exam formula sheet.

Example: 2x² + 7x + 3 = 0

• a = 2, b = 7, c = 3
• x = [-7 ± √(49 - 24)] / 4
• x = [-7 ± √25] / 4
• x = [-7 ± 5] / 4
• x = -0.5 or x = -3

Simultaneous equations

Elimination method:

1. Make coefficients of one variable equal (multiply if needed)
2. Add or subtract equations to eliminate that variable
3. Solve for remaining variable
4. Substitute back to find the other

Example:

• 3x + 2y = 16
• 2x + y = 9

Multiply second equation by 2: 4x + 2y = 18

Subtract first from this: x = 2

Substitute into second: 4 + y = 9, so y = 5

Substitution method:

1. Rearrange one equation to make one variable the subject
2. Substitute into the other equation
3. Solve
4. Find the other variable

Higher tier students must also solve one linear and one quadratic equation simultaneously, typically by substitution.

Sequences

Linear sequences (arithmetic):

Each term increases by a constant difference (common difference, d).

nth term formula: an + b where a is the common difference

To find nth term:

1. Find the common difference (a)
2. Find the zero term (subtract a from first term)
3. Write formula as a × n + b

Example: 5, 8, 11, 14...

• Common difference = 3
• Zero term = 5 - 3 = 2
• nth term = 3n + 2

Quadratic sequences (Higher tier):

Second differences are constant.

nth term has form an² + bn + c

Working method:

1. Find first differences
2. Find second differences
3. a = (second difference) ÷ 2
4. Subtract the an² sequence from original
5. Find nth term of remainder (linear)

Special sequences:

Be familiar with:

• Square numbers: 1, 4, 9, 16, 25...
• Cube numbers: 1, 8, 27, 64...
• Triangle numbers: 1, 3, 6, 10, 15...
• Fibonacci: 1, 1, 2, 3, 5, 8...

Inequalities

Solving linear inequalities:

Apply the same rules as equations, but reverse the inequality sign when multiplying or dividing by a negative number.

Examples:

• 3x + 5 > 17 → 3x > 12 → x > 4
• 5 - 2x ≤ 11 → -2x ≤ 6 → x ≥ -3 (sign reversed)

Representing solutions:

On a number line:

• Open circle (○) for < or >
• Closed circle (●) for ≤ or ≥

Quadratic inequalities (Higher tier):

Sketch the graph or use critical values to determine solution regions.

Rearranging formulae

Make a different variable the subject by applying inverse operations.

Simple rearrangement:

• y = 3x + 5, make x the subject:
• y - 5 = 3x
• x = (y - 5)/3

When subject appears twice:

Collect all terms containing the required subject on one side, factorise, then divide.

Example: Make x the subject of ax + b = cx + d

• ax - cx = d - b
• x(a - c) = d - b
• x = (d - b)/(a - c)

Graphs of functions

Linear graphs:

Form y = mx + c where m is gradient, c is y-intercept.

Gradient = change in y ÷ change in x = (y₂ - y₁)/(x₂ - x₁)

Parallel lines have equal gradients.

Perpendicular lines have gradients that multiply to give -1 (m₁ × m₂ = -1).

Quadratic graphs:

Form y = ax² + bx + c creates a U-shaped (or ∩-shaped) parabola.

Key features:

• Turning point (minimum or maximum)
• Line of symmetry
• y-intercept (when x = 0)
• Roots (where y = 0)

Other graphs (Higher tier):

Recognise shapes of:

• Cubic: y = x³
• Reciprocal: y = 1/x
• Exponential: y = aˣ
• Circular: x² + y² = r²

Worked examples

Example 1: Factorising and solving a quadratic equation (Higher tier, 4 marks)

Question: Solve x² - 3x - 18 = 0

Solution:

Factorise: Find two numbers that multiply to -18 and add to -3

These are -6 and +3

x² - 3x - 18 = (x - 6)(x + 3) = 0 ✓

Set each bracket equal to zero:

x - 6 = 0 or x + 3 = 0 ✓

x = 6 or x = -3 ✓✓

Example 2: Simultaneous equations (Foundation/Higher tier, 4 marks)

Question: Solve the simultaneous equations:

• 2x + 3y = 13
• 3x - y = 3

Solution:

Multiply second equation by 3: 9x - 3y = 9 ✓

Add to first equation: 2x + 3y = 13 9x - 3y = 9 11x = 22 ✓

x = 2 ✓

Substitute into 3x - y = 3: 6 - y = 3 y = 3 ✓

Example 3: nth term of a linear sequence (Foundation/Higher tier, 2 marks)

Question: Find the nth term of the sequence: 11, 17, 23, 29...

Solution:

Common difference = 17 - 11 = 6 ✓

Zero term = 11 - 6 = 5

nth term = 6n + 5 ✓

Common mistakes and how to avoid them

• Sign errors when expanding negative brackets: -3(x - 4) = -3x + 12, not -3x - 12. Multiply the negative by every term inside.

• Not reversing inequality signs when multiplying/dividing by negatives: When solving -2x < 6, dividing by -2 gives x > -3, not x < -3.

• Forgetting to write two solutions for quadratics: A quadratic equation usually has two solutions. After factorising (x - 2)(x + 5) = 0, write both x = 2 AND x = -5.

• Incorrectly combining unlike terms: You cannot simplify 3x + 4x² because the powers are different. Only like terms (same variable and power) can be collected.

• Errors in the quadratic formula: When b is negative, -b becomes positive. For x² - 5x + 6 = 0, use -(-5) = +5 in the formula.

• Not rearranging to = 0 before factorising quadratics: To solve x² + 3x = 10 by factorising, first rearrange to x² + 3x - 10 = 0.

Exam technique for Algebra

• "Solve" means find numerical values: Show all working steps. For linear equations (1-2 marks), for quadratics (3-4 marks), always give exact values unless told to round.

• "Simplify" or "expand and simplify": Collect all like terms completely. An answer with uncollected terms (e.g. 3x + 2x) loses marks even if otherwise correct.

• "Factorise fully": Always check for common factors first. If asked to factorise fully, you must extract all common factors (e.g. 2x² + 8x = 2x(x + 4), not just x(2x + 8)).

• Show substitution clearly: When checking solutions or using formulae, write the substitution line showing numbers replacing letters before calculating. This earns method marks even if arithmetic errors occur.

Quick revision summary

Algebra requires systematic application of rules: collect like terms by combining identical variable parts; expand brackets by multiplying every term; factorise by reversing this process. Solve equations through inverse operations maintaining balance. Linear equations need one solution; quadratics typically need two, found by factorising or using the formula. Sequences have nth term formulae: linear (an + b) or quadratic (an² + bn + c). Inequalities follow equation rules but reverse signs when multiplying/dividing by negatives. Rearrange formulae using inverse operations. Master these methods with careful attention to signs and you'll access marks across all papers.