AQA GCSE Maths: Algebra Revision Guide

Algebra is the largest topic area in AQA GCSE Maths, accounting for approximately 30% of the marks on the Higher tier and around 20% on Foundation. It is tested across all three exam papers -- Paper 1 (non-calculator), Paper 2 (calculator), and Paper 3 (calculator) -- so there is no avoiding it. Many students find algebra the most challenging part of the specification, but it is also one of the most predictable areas to score marks once you have built a systematic approach. The question types repeat year after year, the methods are well-defined, and the mark schemes reward clear, step-by-step working.

This guide covers every major algebra topic on the AQA specification, from basic expressions through to algebraic proof. Each section explains what you need to know, highlights the common pitfalls drawn from examiner reports, and points you toward practice resources. Whether you are aiming for a grade 5 on Foundation or pushing for a grade 9 on Higher, this is your roadmap through the algebra content.

Expressions and Formulae

Algebraic expressions and formulae form the foundation of everything else in algebra. If you cannot simplify, expand, and factorise confidently, you will struggle with equations, graphs, and proof.

Simplifying Expressions

Simplifying means collecting like terms. Terms are "like" if they have the same variable raised to the same power. So 3x and 5x are like terms, but 3x and 3x squared are not. When simplifying, add or subtract the coefficients of like terms: 4x + 7x - 2x = 9x. For expressions with multiple variables, deal with each variable separately: 3a + 5b - 2a + b = a + 6b.

Expanding Brackets

Expanding a single bracket means multiplying every term inside the bracket by the term outside. For example, 3(2x + 5) = 6x + 15. Watch the signs carefully -- a negative multiplier changes the sign of every term inside.

Expanding double brackets requires you to multiply every term in the first bracket by every term in the second bracket. You can use FOIL (First, Outer, Inner, Last) or a grid method. For example, (x + 4)(x - 3) = x squared + 4x - 3x - 12 = x squared + x - 12. Make sure you generate all four terms before simplifying.

Factorising

Factorising is the reverse of expanding. There are several types you need to master:

• Common factor. Look for the highest common factor of all terms and take it outside a bracket. 6x + 15 = 3(2x + 5). For expressions like 4x squared + 10x, the common factor is 2x, giving 2x(2x + 5).
• Difference of two squares (Higher). An expression of the form a squared minus b squared factorises to (a + b)(a - b). For example, x squared - 49 = (x + 7)(x - 7).
• Quadratic trinomials. For expressions of the form x squared + bx + c, find two numbers that multiply to give c and add to give b. For example, x squared + 5x + 6 = (x + 2)(x + 3) because 2 times 3 = 6 and 2 + 3 = 5.
• Quadratic trinomials where the coefficient of x squared is not 1 (Higher). For expressions like 2x squared + 7x + 3, use trial and improvement, grouping, or the AC method. This factorises to (2x + 1)(x + 3).

Substitution into Formulae

Substitution means replacing variables with given numerical values and calculating the result. The key is to work carefully with order of operations (BIDMAS) and to use brackets when substituting negative values. If v = u + at, and u = 3, a = -2, t = 5, then v = 3 + (-2)(5) = 3 + (-10) = -7. Writing the brackets around -2 prevents sign errors.

Rearranging Formulae

Rearranging means changing the subject of a formula. Use inverse operations to isolate the desired variable. To make r the subject of A = pi r squared: divide both sides by pi to get r squared = A / pi, then square root both sides to get r = the square root of (A / pi).

On Higher tier, you need to rearrange formulae where the subject appears twice. For example, making x the subject of y = (3x + 1) / (x - 2). The technique is to multiply out, collect terms containing x on one side, factor out x, and then divide. These questions carry 4-5 marks and reward methodical working.

Equations and Inequalities

Solving equations and inequalities is the most frequently tested skill in GCSE algebra. These questions appear on every paper at every tier.

Solving Linear Equations

A linear equation contains no powers of x higher than 1. The method is to isolate x using inverse operations, performing the same operation on both sides. For 3x + 7 = 22: subtract 7 from both sides (3x = 15), then divide both sides by 3 (x = 5).

Equations with brackets require expanding first. For 4(2x - 3) = 20: expand to get 8x - 12 = 20, add 12 (8x = 32), divide by 8 (x = 4).

Equations with fractions require multiplying through by the denominator. For (x + 5) / 3 = 7: multiply both sides by 3 to get x + 5 = 21, then x = 16. When there are multiple fractions with different denominators, multiply through by the lowest common denominator to clear all fractions at once.

Solving Quadratic Equations

Quadratic equations contain an x squared term and are set equal to zero before solving. There are three methods on the AQA specification.

Factorising. Rearrange the equation so that one side equals zero, factorise, and set each bracket equal to zero. For x squared + 5x + 6 = 0: factorise to (x + 2)(x + 3) = 0, so x = -2 or x = -3. Always give both solutions.

The quadratic formula (Higher). For ax squared + bx + c = 0, the solutions are x = (-b plus or minus the square root of (b squared - 4ac)) / 2a. This works for any quadratic, including those that do not factorise neatly. Substitute the values of a, b, and c carefully, paying close attention to signs. On calculator papers, use the ANS button to avoid retyping long expressions.

Completing the square (Higher). Rewrite x squared + bx + c in the form (x + p) squared + q, where p = b/2 and q = c - p squared. This method is particularly useful for finding the turning point of a quadratic graph and for solving equations where the question specifically requests completing the square.

Simultaneous Equations

Simultaneous equations involve two equations with two unknowns and require you to find values that satisfy both equations at the same time.

Elimination. Multiply one or both equations so that the coefficients of one variable match, then add or subtract the equations to eliminate that variable. Solve for the remaining variable and substitute back to find the other. For example, to solve 2x + 3y = 12 and 5x - 3y = 9: the y coefficients are already matched (3 and -3), so add the equations to get 7x = 21, giving x = 3, then y = 2.

Substitution. Rearrange one equation to express one variable in terms of the other, then substitute into the second equation. This is particularly useful when one equation is already in a convenient form, such as y = 2x + 1.

One linear, one quadratic (Higher). When one equation is linear and the other is quadratic (for example, y = x + 3 and y = x squared + 1), substitute the linear expression into the quadratic and solve the resulting quadratic equation. This typically gives two pairs of solutions.

Linear Inequalities

Solving linear inequalities uses the same method as solving linear equations, with one critical difference: if you multiply or divide both sides by a negative number, you must reverse the inequality sign. For -2x > 6: divide by -2 and flip the sign, giving x < -3.

You also need to represent solutions on a number line. An open circle means the boundary value is not included (for strict inequalities, < or >), and a filled circle means it is included (for non-strict inequalities, less than or equal to, greater than or equal to).

Quadratic Inequalities (Higher)

Solve quadratic inequalities by first solving the corresponding equation to find the critical values, then sketching the quadratic to determine which region satisfies the inequality. For x squared - 5x + 6 < 0: solve x squared - 5x + 6 = 0 to get x = 2 and x = 3, sketch the parabola, and identify that the expression is negative between the roots, giving 2 < x < 3.

Sequences

Sequences appear at both Foundation and Higher tier, and the questions are among the most accessible in algebra if you understand the patterns.

Term-to-Term and Position-to-Term Rules

A term-to-term rule describes how to get from one term to the next. For the sequence 3, 7, 11, 15, ..., the term-to-term rule is "add 4." A position-to-term rule gives the value of any term based on its position in the sequence. For this same sequence, the position-to-term rule (the nth term) is 4n - 1.

Nth Term of Linear Sequences

A linear (arithmetic) sequence has a constant difference between consecutive terms. To find the nth term:

1. Find the common difference -- this is the coefficient of n.
2. Work out what you need to add or subtract to get from the multiples of the common difference to the actual sequence terms.

For the sequence 5, 8, 11, 14, ...: the common difference is 3, so the nth term starts with 3n. The first term is 5 and 3 times 1 is 3, so you add 2. The nth term is 3n + 2. Check: when n = 1, 3(1) + 2 = 5. When n = 2, 3(2) + 2 = 8. Correct.

Nth Term of Quadratic Sequences (Higher)

A quadratic sequence has a constant second difference. The nth term takes the form an squared + bn + c. To find the coefficients:

1. The second difference divided by 2 gives the value of a.
2. Subtract the an squared terms from the original sequence to get a linear sequence.
3. Find the nth term of this linear sequence to determine b and c.

For the sequence 3, 10, 21, 36, 55, ...: the first differences are 7, 11, 15, 19 and the second differences are 4, 4, 4. So a = 4/2 = 2. Subtracting 2n squared (2, 8, 18, 32, 50) from the original sequence gives 1, 2, 3, 4, 5 -- which has nth term n. So the nth term is 2n squared + n.

Fibonacci-Type and Geometric Sequences

Fibonacci-type sequences add the two previous terms to get the next term. You will not be asked for the nth term of these, but you need to recognise and continue them.

Geometric sequences (Higher) have a constant ratio between consecutive terms. The sequence 2, 6, 18, 54, ... is geometric with common ratio 3. You need to identify whether a sequence is arithmetic, geometric, or neither, and continue it.

Graphs

Graph work connects algebra to visual representation and is tested heavily across all three papers.

Plotting Linear Graphs

The equation y = mx + c defines a straight line where m is the gradient and c is the y-intercept. To plot a straight line, you can either create a table of values and plot points, or use the gradient and y-intercept directly. For y = 2x + 3: the y-intercept is (0, 3) and the gradient is 2, meaning the line rises 2 units for every 1 unit across.

Gradient and Y-Intercept

The gradient of a line measures its steepness. A positive gradient slopes upward from left to right; a negative gradient slopes downward. To find the gradient from a graph, choose two points on the line and calculate the change in y divided by the change in x.

The y-intercept is the point where the line crosses the y-axis, read directly from the graph or from the constant term in y = mx + c.

Parallel and Perpendicular Lines

Parallel lines have the same gradient. If one line has equation y = 3x + 1 and another line is parallel to it, the second line also has gradient 3.

Perpendicular lines (Higher) have gradients that are negative reciprocals of each other. If one line has gradient 3, a perpendicular line has gradient -1/3. The product of the two gradients is -1.

Quadratic and Cubic Graphs

Quadratic graphs (y = ax squared + bx + c) produce a parabola. If the coefficient of x squared is positive, the parabola opens upward; if negative, it opens downward. The turning point (vertex) can be found by completing the square.

Cubic graphs (y = ax cubed + ...) produce an S-shaped curve. You need to be able to plot these from a table of values and to recognise their general shape.

Real-Life Graphs

Distance-time graphs show journeys. The gradient represents speed -- a steeper line means faster travel, a horizontal line means stationary, and a line sloping back toward zero means returning.

Speed-time graphs show how speed changes over time. The gradient represents acceleration. The area under a speed-time graph represents the total distance travelled.

Areas Under Graphs (Higher)

Estimating the area under a curve using the trapezium rule is a Higher tier skill. Divide the area into strips, treat each strip as a trapezium, calculate the area of each, and sum them.

Transformations of Graphs (Higher)

You need to know the effect of the following transformations:

• y = f(x) + a translates the graph a units up.
• y = f(x + a) translates the graph a units to the left.
• y = -f(x) reflects the graph in the x-axis.
• y = f(-x) reflects the graph in the y-axis.

Functions and Iteration (Higher)

Functions and iteration are Higher tier topics that appear in the later questions on each paper.

Function Notation

f(x) is simply another way of writing a formula. If f(x) = 3x + 2, then f(5) = 3(5) + 2 = 17. You substitute the input value wherever you see x.

Composite Functions

A composite function applies one function followed by another. fg(x) means "apply g first, then apply f to the result." If f(x) = 2x + 1 and g(x) = x squared, then fg(3) = f(g(3)) = f(9) = 2(9) + 1 = 19. The order matters -- fg(x) is not the same as gf(x).

Inverse Functions

The inverse function f inverse (written f to the power of -1) reverses what f does. To find it, write y = f(x), swap x and y, and rearrange for y. If f(x) = 2x + 5, then y = 2x + 5 becomes x = 2y + 5, which rearranges to y = (x - 5) / 2. So f inverse of x = (x - 5) / 2.

Iteration

Iteration uses a formula to generate a sequence of approximations that converge toward the solution of an equation. Given a starting value x0 and an iteration formula such as x(n+1) = (x(n) squared + 3) / 5, you substitute x0 to find x1, then x1 to find x2, and so on. The values converge toward the solution. Show each iteration clearly and give your final answer to the required degree of accuracy.

Algebraic Proof

Algebraic proof is a Higher tier topic that many students find intimidating, but the question types are highly predictable and the technique is learnable.

Structuring a Proof

A proof starts from a clearly defined set of assumptions and proceeds through logical steps to reach the required conclusion. Every step must follow from the previous one, and you cannot assume what you are trying to prove.

Common Proof Types

Even and odd proofs. Represent any integer as n. Then 2n is even and 2n + 1 is odd. To prove that the sum of two odd numbers is always even: (2n + 1) + (2m + 1) = 2n + 2m + 2 = 2(n + m + 1), which is a multiple of 2 and therefore even.

Consecutive integer proofs. Represent consecutive integers as n, n + 1, n + 2, and so on. To prove that the sum of three consecutive integers is always a multiple of 3: n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1), which is a multiple of 3.

Divisibility proofs. Show that an expression is always divisible by a given number by factorising it so that the given number appears as a factor.

Showing that an expression is always positive. Complete the square to show the expression can be written as a perfect square plus a positive constant, which must always be positive.

The key to proof questions is practice. Learn the standard openings (let n be any integer, let 2n be any even number, let 2n + 1 be any odd number) and practise the common patterns until they feel routine.

Common Mistakes in Algebra

Examiner reports highlight the same algebra errors year after year. Knowing them helps you avoid them.

Sign errors when expanding brackets. -3(x - 4) = -3x + 12, not -3x - 12. The most common single error in GCSE algebra. When you multiply a negative by a negative, the result is positive.

Forgetting to multiply all terms. When expanding 2(3x + 4y - 5), every term inside the bracket must be multiplied by 2: 6x + 8y - 10. Students often multiply the first term and forget the rest.

Incorrect factorising. Always check your factorisation by expanding back out. If (x + 3)(x - 2) does not expand to give you the original expression, something is wrong.

Not showing sufficient working in "show that" questions. If the question says "show that x squared + 6x + 9 = (x + 3) squared," you must expand (x + 3) squared step by step. Writing only the answer earns zero marks because the answer was given in the question.

Misreading inequality signs. The difference between < and less than or equal to matters. On a number line, an open circle means the value is not included; a filled circle means it is.

Errors with negative coefficients in simultaneous equations. When eliminating variables, students frequently make sign errors, particularly when subtracting an equation that contains negative terms. Write out the subtraction term by term and double-check each sign.

Forgetting plus-or-minus when solving quadratics. If x squared = 16, then x = 4 or x = -4. Missing the negative root loses marks.

Exam Technique for Algebra

Algebra appears on all three papers, so you need to be prepared for both calculator and non-calculator algebra questions.

Paper 1 (Non-Calculator)

On the non-calculator paper, you need to solve equations entirely by hand. The numbers will be chosen to work out neatly, but you must be careful with arithmetic -- particularly with negative numbers and fractions. Show every step of your working. If you are solving a quadratic by factorising, write out the factorisation, set each bracket equal to zero, and state both solutions separately.

Papers 2 and 3 (Calculator)

On calculator papers, the quadratic formula becomes more practical because the arithmetic can be messy. Use your calculator carefully, entering the full expression in one go if possible. Check that you have entered the formula correctly by verifying the discriminant (b squared - 4ac) separately.

General Advice

Show every step. Method marks are available on questions worth 2 marks or more. A correct method with an arithmetic slip earns most of the marks; a correct answer with no visible working earns nothing if the answer is wrong.

Check solutions by substituting back. After solving an equation, substitute your answer back into the original equation to verify it works. This takes seconds and catches errors.

For graph questions, use a ruler and plot points accurately. Marks are awarded for accuracy of plotting. A wobbly line or points that are even half a square out can cost marks. Use a sharp pencil and a straight edge.

Read the question carefully. Does it ask for the value of x, or the value of 2x + 1? Does it ask you to factorise, or to solve? The method may be the same, but the final answer is different.

On "show that" questions, work forwards from the given information, not backwards from the answer. The answer is already in the question -- your job is to demonstrate the logical steps that lead to it.

Prepare with LearningBro

LearningBro's GCSE Mathematics Algebra course covers every algebra topic on the AQA specification with practice questions that mirror the format and difficulty of the real exam. Work through expressions, equations, inequalities, sequences, graphs, and proof with immediate feedback on your answers.

For broader exam preparation covering all topic areas and paper strategies, try our GCSE Mathematics AQA Exam Prep course. And for a detailed breakdown of how to approach each of the three papers, read our AQA GCSE Maths exam technique guide.

Algebra is a skill built through practice. The more equations you solve, the more graphs you plot, and the more proofs you write, the more automatic the methods become. Start with the topics you find hardest, work through them systematically, and you will find that 30% of the exam becomes one of the most reliable areas for picking up marks.

Good luck with your revision.