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Edexcel IGCSE Maths A (4MA1) · Foundation tier · Revision notes

IGCSE Maths 4MA1 Foundation Algebra Notes

Revision notes for equations, formulae and identities at Foundation tier (Papers 1F and 2F, grades 5 to 1). Each sub-topic has what you need to know, the key methods, a common mistake and a worked example. Cover the worked solution and try the example first.

Algebra questions with answers →

2.1 Use of symbols

What you need to know: letters for numbers and variables; simplifying expressions; index laws in algebra, including zero and negative powers.

  • A letter can stand for a number in an expression or a variable in a formula; 3a means 3 × a.
  • Collect like terms by adding or subtracting the terms with exactly the same letters and powers.
  • Index laws in algebra: x³ × x⁴ = x⁷, x⁹ ÷ x² = x⁷ and (x³)² = x⁶.
  • Zero and negative powers follow the same laws: x⁰ = 1 and x⁻² = 1/x².

Watch out: x² and x are not like terms, so x² + x cannot be simplified to x³.

Worked example (2 marks, grade 3):

Write as a single power of \(x\)

(a) \(x^5 \div x^8\)

(b) \(x^4 \div x^4\)

  1. (a) 5 − 8 = −3, so x⁻³.
  2. (b) 4 − 4 = 0, so x⁰, which equals 1.

Answer: (a) \(x^{-3}\) (b) \(x^0 = 1\)

Higher tier only: fractional powers.

Practise: Use of symbols questions with answers · Unit 2 practice (Foundation) · Higher level: Indices and Standard Form questions · Algebraic Expressions questions

2.2 Algebraic manipulation

What you need to know: substitution; expanding single and double brackets; factorising with common factors; factorising x² + bx + c.

  • Expand a single bracket by multiplying every term inside by the term outside.
  • Expand two brackets by multiplying each term in the first by each term in the second (four products), then collect like terms.
  • Factorise by taking out the highest common factor of every term.
  • To factorise x² + bx + c, find two numbers that multiply to c and add to b.

Watch out: −3(x − 4) is −3x + 12: the minus sign multiplies both terms.

Worked example (2 marks, grade 3):

Factorise fully \(10ab + 15b\)

  1. The highest common factor of 10ab and 15b is 5b.
  2. 10ab ÷ 5b = 2a and 15b ÷ 5b = 3.

Answer: \(5b(2a + 3)\)

Higher tier only: three or more brackets, factorising ax² + bx + c, algebraic fractions, completing the square and algebraic proof.

Practise: Algebraic manipulation questions with answers · Unit 2 practice (Foundation) · Unit 3 practice (Foundation) · Higher level: Expanding and Factorising questions · Quadratic Equations questions

2.3 Expressions and formulae

What you need to know: writing expressions and formulae from words and diagrams; substituting into formulae; changing the subject when it appears once.

  • Write a formula from words by turning each part of the rule into symbols, for example a fixed charge plus a cost per mile.
  • To substitute, replace each letter by its value in brackets, then follow the order of operations.
  • To change the subject, undo what has been done to the new subject in reverse order, doing the same to both sides.
  • When the subject is squared, rearrange for the square first and then take the square root.

Watch out: With negative values, use brackets: 3 × (−2) is −6.

Worked example (2 marks, grade 3):

Make \(x\) the subject of \(y = 4x - 7\)

  1. Add 7 to both sides: y + 7 = 4x.
  2. Divide both sides by 4: x = (y + 7)/4.

Answer: \(x = \frac{y + 7}{4}\)

Higher tier only: changing the subject when it appears twice or as a power.

Practise: Expressions and formulae questions with answers · Unit 2 practice (Foundation) · Higher level: Algebraic Expressions questions · Rearranging Formulae questions

2.4 Linear equations

What you need to know: solving linear equations, including brackets and fractions; forming equations from information.

  • Solve an equation by doing the same to both sides until the unknown is on its own.
  • Expand brackets first, then collect the unknown on the side with the larger number of it.
  • With fractions, multiply every term by the lowest common denominator to clear them.
  • To form an equation, write an expression for each quantity and set them equal, for example angles in a triangle add up to 180°.

Watch out: When you multiply both sides to clear a fraction, multiply every term, not just one.

Worked example (2 marks, grade 3):

Solve \(7x - 4 = 3x + 18\)

  1. Subtract 3x from both sides: 4x − 4 = 18.
  2. Add 4: 4x = 22.
  3. Divide by 4: x = 5.5.

Answer: \(x = 5.5\)

Practise: Linear equations questions with answers · Unit 3 practice (Foundation) · Higher level: Linear Equations questions

2.6 Simultaneous linear equations

What you need to know: solving two simultaneous linear equations exactly.

  • Simultaneous equations have one pair of values that makes both equations true.
  • Elimination: make the number in front of one letter the same in both equations, then add or subtract to remove it.
  • Substitution: if one equation gives y in terms of x, put that expression into the other equation.
  • Substitute your first answer back into an equation to find the other letter, and check in the second equation.

Watch out: Subtract when the signs are the same and add when they are different.

Worked example (2 marks, grade 3):

Solve the simultaneous equations

\(x + y = 11\)

\(x - y = 3\)

  1. Add the equations: 2x = 14, so x = 7.
  2. Substitute: 7 + y = 11, so y = 4.

Answer: \(x = 7\), \(y = 4\)

Higher tier only: interpreting the solution as the point where two lines meet.

Practise: Simultaneous linear equations questions with answers · Unit 3 practice (Foundation) · Higher level: Simultaneous Equations questions

2.7 Quadratic equations

What you need to know: solving quadratic equations x² + bx + c = 0 by factorising.

  • Rearrange so that one side is zero, then factorise the quadratic x² + bx + c.
  • If two brackets multiply to give zero, one of them must be zero, which gives the two solutions.
  • A quadratic can have two solutions, one repeated solution, as in (x − 6)² = 0, or two that are equal and opposite, as in x² = 49.
  • In a context question, reject a solution that makes no sense, such as a negative length.

Watch out: If (x + 4)(x − 2) = 0 then x = −4 or x = 2: the signs change.

Worked example (2 marks, grade 3):

Solve \(x^2 + 7x + 12 = 0\)

  1. Factorise: (x + 3)(x + 4) = 0.
  2. x + 3 = 0 or x + 4 = 0, so x = −3 or x = −4.

Answer: \(x = -3\) or \(x = -4\)

Higher tier only: the quadratic formula, completing the square, forming quadratics from a context and linear–quadratic simultaneous equations.

Practise: Quadratic equations questions with answers · Unit 3 practice (Foundation) · Higher level: Quadratic Equations questions

2.8 Inequalities

What you need to know: inequality symbols and number lines; solving linear inequalities; regions on a coordinate grid.

  • < and > are strict; ≤ and ≥ include the end value.
  • On a number line an open circle means the value is not included and a filled circle means it is.
  • Solve an inequality like an equation, but if you multiply or divide by a negative number, reverse the sign.
  • On a grid, draw each boundary line and find the region where all the inequalities are true; test a point to check the side.

Watch out: −3 ≤ n < 2 includes −3 but not 2.

Worked example (2 marks, grade 3):

Solve \(5x - 7 \ge 2x + 8\)

  1. Subtract 2x: 3x − 7 ≥ 8.
  2. Add 7: 3x ≥ 15, so x ≥ 5.

Answer: \(x \ge 5\)

Higher tier only: quadratic inequalities and harder regions.

Practise: Inequalities questions with answers · Unit 2 practice (Foundation) · Unit 4 practice (Foundation) · Higher level: Inequalities questions · Inequalities on Graphs questions

Topic list for Edexcel IGCSE Maths 4MA1 Foundation algebra: use of symbols, algebraic manipulation, expressions and formulae, linear equations, simultaneous linear equations, quadratic equations, inequalities
The 7 Foundation sub-topics of algebra.

Practise algebra