Algebra

What you'll learn

Algebra forms approximately 30% of your OCR GCSE Mathematics papers and underpins much of the remaining content. This guide covers all algebraic skills required for both Foundation and Higher tiers, from basic manipulation through to complex problem-solving with simultaneous equations and quadratics. You'll develop fluency in algebraic notation, learn systematic approaches to solving equations, and understand how to interpret and manipulate algebraic expressions confidently.

Key terms and definitions

Variable — a symbol (usually a letter) that represents an unknown or changeable quantity

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

Expression — a mathematical statement containing variables, numbers and operations but no equals sign (e.g., 3x + 7)

Term — a single number or variable, or numbers and variables multiplied together, separated by + or − signs

Linear equation — an equation where the highest power of the variable is 1 (e.g., 2x + 5 = 13)

Quadratic expression — an expression where the highest power of the variable is 2 (e.g., x² + 3x − 10)

Factorising — writing an expression as a product of its factors

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

Core concepts

Simplifying and manipulating expressions

Collecting like terms involves combining terms with identical variable parts:

• 5x + 3y − 2x + 7y = 3x + 10y
• Only combine terms with the same variables raised to the same powers
• 4x² and 3x are not like terms

Expanding brackets requires multiplying everything inside the bracket by the term outside:

• Single brackets: 4(2x + 5) = 8x + 20
• Two brackets: (x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15
• Use FOIL (First, Outer, Inner, Last) or the grid method for double brackets

Factorising reverses expansion:

• Common factors: 6x + 15 = 3(2x + 5)
• Quadratics: x² + 7x + 12 = (x + 3)(x + 4)
• Difference of two squares: x² − 25 = (x + 5)(x − 5)

Higher tier students must also handle:

• Expanding three brackets
• Factorising harder quadratics where the x² coefficient ≠ 1
• Algebraic fractions requiring common denominators

Solving linear equations

A systematic approach ensures accuracy:

One-step equations:

• x + 7 = 15 → x = 8 (subtract 7 from both sides)
• 4x = 28 → x = 7 (divide both sides by 4)

Multi-step equations:

1. Expand any brackets
2. Collect all x terms on one side
3. Collect all number terms on the other side
4. Divide to find x

Example: 3(2x − 5) = 4x + 7

• Expand: 6x − 15 = 4x + 7
• Subtract 4x: 2x − 15 = 7
• Add 15: 2x = 22
• Divide by 2: x = 11

Equations with fractions:

• Multiply through by the denominators to clear fractions
• (x + 3)/4 = 5 → x + 3 = 20 → x = 17

Equations with x on both sides:

• Always collect variables on the side where they'll remain positive
• 5x − 3 = 2x + 12 → 3x = 15 → x = 5

Solving quadratic equations

Quadratics can be solved by factorising (when possible), using the quadratic formula, or by completing the square.

Solving by factorising:

1. Rearrange to equal zero
2. Factorise the quadratic
3. Set each bracket equal to zero
4. Solve each linear equation

Example: x² + 5x − 14 = 0

• Factorise: (x + 7)(x − 2) = 0
• Either x + 7 = 0 or x − 2 = 0
• Solutions: x = −7 or x = 2

The quadratic formula (Higher tier): For ax² + bx + c = 0:

x = (−b ± √(b² − 4ac)) / 2a

The discriminant b² − 4ac determines the number of solutions:

• b² − 4ac > 0: two distinct real solutions
• b² − 4ac = 0: one repeated solution
• b² − 4ac < 0: no real solutions

Completing the square (Higher tier):

• Converts x² + bx + c to (x + p)² + q form
• For x² + 6x + 2: take half the x coefficient → (x + 3)² − 9 + 2 = (x + 3)² − 7
• Useful for finding turning points and solving equations

Simultaneous equations

Elimination method (when coefficients can be matched):

1. Multiply one or both equations to make coefficients of one variable match
2. Add or subtract equations to eliminate that variable
3. Solve for the remaining variable
4. Substitute back to find the other variable

Substitution method:

1. Rearrange one equation to make one variable the subject
2. Substitute this expression into the other equation
3. Solve the resulting equation
4. Substitute back to find the other variable

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

Sequences

Linear sequences have a constant difference between consecutive terms:

• nth term formula: an + b
• 'a' is the common difference
• To find b: substitute n = 1 and solve

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

• Common difference = 3
• nth term = 3n + b
• When n = 1: 3(1) + b = 5, so b = 2
• nth term = 3n + 2

Quadratic sequences (Higher tier) have a constant second difference:

• nth term formula contains n²
• Method involves finding the coefficient of n² from half the second difference

Fibonacci-type sequences where each term is generated from previous terms:

• Example: 2, 5, 7, 12, 19... (each term = sum of previous two)

Inequalities

Use the same methods as equations, but remember:

• Multiplying or dividing by a negative number reverses the inequality sign
• Represent solutions on a number line
• Open circle (○) for < or >, closed circle (●) for ≤ or ≥

Solving: 3x − 7 < 8

• Add 7: 3x < 15
• Divide by 3: x < 5

Representing regions (Higher tier):

• Inequalities like y < 2x + 3 define regions on graphs
• Solid line for ≤ or ≥, dashed line for < or >
• Shade the required region

Graphs of algebraic functions

Linear graphs y = mx + c:

• m is the gradient (steepness)
• c is the y-intercept (where the line crosses the y-axis)
• Parallel lines have equal gradients
• Perpendicular lines have gradients that multiply to give −1

Quadratic graphs y = ax² + bx + c:

• U-shaped (a > 0) or ∩-shaped (a < 0) curves called parabolas
• Turning point (minimum or maximum)
• Line of symmetry at x = −b/2a (Higher tier)

Other graphs (Higher tier):

• Cubic graphs (x³): distinctive S-shape
• Reciprocal graphs (y = k/x): two curves in opposite quadrants with asymptotes
• Exponential graphs (y = aˣ): rapid growth or decay

Graph transformations (Higher tier):

• f(x) + a shifts graph up by a units
• f(x + a) shifts graph left by a units
• af(x) stretches graph vertically by factor a
• f(ax) stretches graph horizontally by factor 1/a

Algebraic proof

Higher tier students must construct logical arguments:

Proving statements about numbers:

• Express general cases algebraically
• Show = manipulations leading to required result
• Use phrases: "Let n be an integer..."

Example: Prove the sum of two consecutive odd numbers is always even.

• Let first odd number = 2n + 1
• Next odd number = 2n + 3
• Sum = (2n + 1) + (2n + 3) = 4n + 4 = 2(2n + 2)
• This is 2 × (an integer), so is even

Disproof by counter-example:

• One counter-example disproves a statement
• "n² > n for all values of n" is disproved by n = 0.5 since 0.25 < 0.5

Worked examples

Example 1: Solving equations with fractions (Foundation/Higher)

Question: Solve (2x + 5)/3 = (x − 1)/2

Solution: Multiply both sides by 6 (LCM of 2 and 3):

• 2(2x + 5) = 3(x − 1)
• 4x + 10 = 3x − 3

Collect x terms on left, numbers on right:

• 4x − 3x = −3 − 10
• x = −13

Check: (−26 + 5)/3 = −21/3 = −7 and (−13 − 1)/2 = −14/2 = −7 ✓

Example 2: Simultaneous equations (Higher)

Question: Solve algebraically: y = x² − 4x + 1 y = 2x − 5

Solution: Since both expressions equal y, set them equal:

• x² − 4x + 1 = 2x − 5

Rearrange to standard form:

• x² − 6x + 6 = 0

Use quadratic formula (a = 1, b = −6, c = 6):

• x = (6 ± √(36 − 24))/2 = (6 ± √12)/2 = (6 ± 2√3)/2 = 3 ± √3

Find y values:

• When x = 3 + √3: y = 2(3 + √3) − 5 = 1 + 2√3
• When x = 3 − √3: y = 2(3 − √3) − 5 = 1 − 2√3

Solutions: (3 + √3, 1 + 2√3) and (3 − √3, 1 − 2√3)

Example 3: Algebraic proof (Higher)

Question: Prove that (n + 1)² − (n − 1)² is a multiple of 4 for all positive integers n.

Solution: Expand both brackets:

• (n + 1)² = n² + 2n + 1
• (n − 1)² = n² − 2n + 1

Subtract:

• (n² + 2n + 1) − (n² − 2n + 1)
• = n² + 2n + 1 − n² + 2n − 1
• = 4n

Factor:

• = 4 × n

Since n is an integer, 4n is 4 multiplied by an integer, therefore it is a multiple of 4.

Common mistakes and how to avoid them

• Sign errors when expanding negative brackets: −3(x − 5) = −3x + 15 not −3x − 15. Multiply everything inside by −3.

• Forgetting to perform operations on both sides: If you add 5 to the left side of an equation, you must add 5 to the right side too. Keep equations balanced.

• Confusing expressions and equations: 3x + 7 is an expression (simplify it); 3x + 7 = 22 is an equation (solve it). Know which task you're performing.

• Dividing by variables incorrectly: Never divide both sides by x unless you know x ≠ 0. You might lose solutions.

• Reversing inequality signs without cause: Only reverse when multiplying or dividing by a negative number, not when simply rearranging.

• Incorrect factorising: Always expand your factorised answer to check it matches the original expression. For x² + 7x + 12, check that (x + 3)(x + 4) = x² + 7x + 12.

Exam technique for Algebra

• "Solve" means find the value(s) of the variable. Always give your final answer as x = ... or y = ... Show clear working for method marks even if your answer is incorrect.

• "Simplify" requires collecting like terms or cancelling, not solving. Your answer will still contain variables. Do not introduce equals signs where none exist.

• "Show that" or "Prove" questions require complete mathematical reasoning. You must demonstrate every step logically. Simply verifying with numbers earns no marks. State conclusions clearly.

• Mark allocation indicates required detail: 1 mark = simple step or final answer; 3-4 marks = multiple steps required. If a 3-mark question seems easy, you've likely missed steps.

Quick revision summary

Algebra requires systematic methods applied consistently. Master simplifying expressions by collecting like terms and expanding brackets. Solve linear equations by isolating the variable through inverse operations. Factorise quadratics to solve them, or use the quadratic formula for difficult cases. Simultaneous equations need elimination or substitution methods. Sequences have formulae linking position to value. Inequalities follow equation rules except when multiplying or dividing by negatives. Sketch graphs by identifying key features. Higher tier demands proof through algebraic manipulation and logical reasoning.