Algebra

What you'll learn

Algebra forms approximately 30% of your Pearson Edexcel International IGCSE Mathematics examination. This guide covers all testable algebraic content including manipulation of expressions, solving equations and inequalities, sequences, graphs, and functions. You'll develop fluency in algebraic techniques essential for both Foundation and Higher tier papers.

Key terms and definitions

Variable — a symbol (usually a letter) that represents an unknown quantity or a quantity that can take different values

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

Equation — a mathematical statement showing two expressions are equal, containing an equals sign that can be solved to find unknown values

Coefficient — the number multiplying a variable in an algebraic term (e.g., in 5x², the coefficient is 5)

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

Quadratic equation — an equation containing a term in x² as the highest power, typically written as ax² + bx + c = 0

Simultaneous equations — two or more equations with multiple unknowns that are solved together to find values satisfying all equations

Factorising — writing an expression as a product of its factors, the reverse process of expanding brackets

Core concepts

Algebraic manipulation

Simplifying expressions involves collecting like terms and performing operations systematically.

When expanding brackets, multiply each term inside the bracket by the term outside:

• Single brackets: 3(2x + 5) = 6x + 15
• Double brackets: (x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15

Factorising is the inverse of expansion. Common methods include:

Single factor extraction:

• Identify the highest common factor (HCF) of all terms
• Example: 6x² + 9x = 3x(2x + 3)

Quadratic expressions:

• For x² + bx + c, find two numbers that multiply to c and add to b
• Example: x² + 7x + 12 = (x + 3)(x + 4)

Difference of two squares:

• Pattern: a² - b² = (a + b)(a - b)
• Example: x² - 16 = (x + 4)(x - 4)

Algebraic fractions follow the same rules as numerical fractions:

• Simplify by cancelling common factors
• Add/subtract: find a common denominator
• Multiply: multiply numerators, multiply denominators
• Divide: multiply by the reciprocal

Solving equations

Linear equations require isolating the variable through inverse operations:

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

Example: 3(x - 2) = 15

• 3x - 6 = 15
• 3x = 21
• x = 7

Quadratic equations can be solved by:

Factorising:

• Rearrange to form ax² + bx + c = 0
• Factorise the quadratic
• Set each factor equal to zero
• Example: x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0 → x = -2 or x = -3

Quadratic formula: For ax² + bx + c = 0: x = [-b ± √(b² - 4ac)] / 2a

The discriminant (b² - 4ac) determines the number of solutions:

• Positive: two distinct real solutions
• Zero: one repeated solution
• Negative: no real solutions

Completing the square:

• Write x² + bx + c in the form (x + p)² + q
• Example: x² + 6x + 2 = (x + 3)² - 9 + 2 = (x + 3)² - 7

Simultaneous equations involve two unknowns:

Elimination method:

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

Substitution method:

1. Rearrange one equation for one variable
2. Substitute into the other equation
3. Solve, then substitute back

Sequences and functions

Sequences are ordered lists of numbers following a pattern.

The nth term formula allows you to find any term without listing all previous terms:

Arithmetic sequences (linear) have a constant difference:

• General form: nth term = dn + (a - d)
• Where d = common difference, a = first term

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

• Difference = 3
• nth term = 3n + 2

Geometric sequences have a constant ratio:

• Each term = previous term × common ratio

Functions describe relationships between inputs and outputs, denoted f(x).

Function notation:

• f(x) = 2x + 3 means "the function f applied to x equals 2x + 3"
• f(5) means substitute x = 5 into the function

Composite functions:

• fg(x) means apply g first, then apply f to the result
• Example: if f(x) = 2x and g(x) = x + 3, then fg(x) = 2(x + 3) = 2x + 6

Inverse functions reverse the original function:

• Denoted f⁻¹(x)
• To find: swap x and y, then solve for y

Inequalities

Inequalities use symbols: < (less than), > (greater than), ≤ (less than or equal), ≥ (greater than or equal)

Solve like equations but remember:

• Multiplying or dividing by a negative number reverses the inequality sign
• Example: -2x > 6 becomes x < -3

Represent solutions on number lines:

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

Quadratic inequalities require finding critical values:

1. Solve the corresponding quadratic equation
2. Sketch the quadratic curve
3. Identify regions satisfying the inequality

Graphs of functions

Linear graphs have the form y = mx + c:

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

Gradient between two points (x₁, y₁) and (x₂, y₂): m = (y₂ - y₁)/(x₂ - x₁)

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

• y = ax² + bx + c
• Vertex (turning point) found by completing the square
• Axis of symmetry at x = -b/2a

Other key graphs:

• Cubic: y = ax³ (S-shaped curve)
• Reciprocal: y = k/x (hyperbola)
• Exponential: y = aˣ (rapid growth/decay)

Transformations of graphs:

• y = f(x) + a: translation a units up
• y = f(x + a): translation a units left
• y = af(x): vertical stretch, scale factor a
• y = f(ax): horizontal stretch, scale factor 1/a
• y = -f(x): reflection in x-axis
• y = f(-x): reflection in y-axis

Interpreting graphs

Distance-time graphs:

• Gradient = speed
• Horizontal line = stationary
• Steeper gradient = faster speed

Speed-time graphs:

• Gradient = acceleration
• Area under graph = distance travelled
• Horizontal line = constant speed

Worked examples

Example 1: Solving simultaneous equations (Higher tier, 4 marks)

Question: Solve the simultaneous equations: 3x + 2y = 16 5x - y = 9

Solution:

Label the equations: 3x + 2y = 16 ... (1) 5x - y = 9 ... (2)

Multiply equation (2) by 2: 10x - 2y = 18 ... (3)

Add equations (1) and (3): 3x + 2y + 10x - 2y = 16 + 18 13x = 34 x = 34/13 = 2.615... [1 mark for valid elimination method]

Wait - check if integer solution expected. Try again: 13x = 34 doesn't give neat answer. Recheck calculation.

Actually: 3x + 2y = 16 and 10x - 2y = 18 Adding: 13x = 34, so x = 34/13

This suggests calculation error. Verify original: If 5x - y = 9, then multiplying by 2: 10x - 2y = 18 ✓

13x = 34 is unusual. But x = 2.615...

Substitute into (2): 5(2.615) - y = 9 Wait — round x sensibly or keep exact.

Better approach: x = 34/13 = 2 8/13 (unusual)

Let's verify the question is correct and solve: From (2): y = 5x - 9 Substitute into (1): 3x + 2(5x - 9) = 16 [1 mark for substitution] 3x + 10x - 18 = 16 13x = 34 x = 34/13 [1 mark for correct x value]

y = 5(34/13) - 9 = 170/13 - 117/13 = 53/13 [1 mark for correct y value]

(Note: Real exam questions typically have integer solutions)

Example 2: Quadratic equation by factorising (Foundation/Higher, 3 marks)

Question: Solve x² - 5x - 14 = 0

Solution:

Factorise: need two numbers that multiply to -14 and add to -5 These are -7 and +2 x² - 5x - 14 = (x - 7)(x + 2) = 0 [2 marks for correct factorisation]

If (x - 7)(x + 2) = 0, then: x - 7 = 0 or x + 2 = 0 x = 7 or x = -2 [1 mark for both solutions]

Example 3: Finding the nth term (Foundation/Higher, 2 marks)

Question: Find the nth term of the sequence: 7, 10, 13, 16, 19...

Solution:

First difference = 10 - 7 = 3 (constant, so linear sequence) [1 mark for identifying arithmetic sequence]

nth term = 3n + c

When n = 1: 3(1) + c = 7, so c = 4

nth term = 3n + 4 [1 mark for correct formula]

Check: when n = 2, 3(2) + 4 = 10 ✓

Common mistakes and how to avoid them

• Forgetting to change the sign when moving terms across the equals sign. When solving 5x + 3 = 18, students write 5x = 18 + 3 instead of 5x = 18 - 3. Always perform the inverse operation.

• Incorrect distribution when expanding brackets. Writing 3(2x + 5) = 6x + 5 instead of 6x + 15. Multiply EVERY term inside the bracket by the term outside.

• Dividing instead of multiplying by the reciprocal for algebraic fractions. To divide by a fraction, multiply by its reciprocal: (x/3) ÷ (2/5) = (x/3) × (5/2).

• Reversing inequality signs incorrectly. Only reverse when multiplying or dividing both sides by a negative number, not when moving terms.

• Finding only one solution to quadratic equations. Quadratics typically have two solutions. After factorising, set BOTH factors equal to zero.

• Confusing function composition order. For fg(x), apply g first, then f. Read from right to left, not left to right.

Exam technique for Algebra

• "Solve" versus "Simplify": "Solve" requires finding numerical values for variables and needs an equals sign. "Simplify" means rewrite in a neater form without solving.

• Show full working for method marks. Even if your final answer is incorrect, you can earn marks for correct methods. Write each step on a new line for clarity.

• Check solutions by substitution. If time permits, substitute your answer back into the original equation to verify correctness. This catches arithmetic errors.

• Use exact values unless told to round. Leave answers as fractions or surds (like √2) unless the question specifies "give your answer to 2 decimal places." Premature rounding loses accuracy marks.

Quick revision summary

Algebra requires systematic manipulation of expressions and equations using inverse operations. Master factorising and expanding brackets for both linear and quadratic expressions. Solve equations by isolating variables, using factorisation, the quadratic formula, or simultaneous equation methods. Recognise arithmetic sequences and derive nth term formulae. Understand function notation, composition, and inverses. Sketch and transform common graph types including linear, quadratic, cubic, reciprocal, and exponential functions. Interpret real-world graphs and apply inequality notation correctly, remembering to reverse signs when multiplying or dividing by negative numbers.