What is algebraic manipulation?
Algebraic manipulation is the process of rearranging, simplifying, and transforming algebraic expressions and equations. It is one of the most fundamental skills in GCSE Maths — appearing in virtually every topic from geometry to statistics.
Mastering algebraic manipulation means you can confidently:
- Simplify and collect like terms
- Substitute values into expressions and formulae
- Expand single and double brackets
- Factorise expressions including quadratics
- Change the subject of a formula
Choose a topic below for full worked examples, graded practice questions and a common mistakes section.
Algebraic manipulation topics
Single brackets, double brackets (FOIL), perfect squares and the difference of two squares. Includes Grade 3–9 practice questions.
Common factor factorising, factorising quadratics (a=1 and a≠1), difference of two squares and grouping. Grade 3–9.
Substituting positive and negative values into expressions and formulae, including with powers and roots. Grade 3–9.
Rearranging formulae to isolate a given variable — including with squares, roots and the subject appearing twice. Grade 4–9.
Simplifying and collecting like terms
Like terms share exactly the same variable(s) raised to exactly the same power. You can only add or subtract terms that are "like". Constants (plain numbers with no variable) are always like terms with each other.
For example: 5x + 3y − 2x + 7y simplifies to 3x + 10y — group the x terms and y terms separately, then combine each group.
A trickier example: 4a² + 3a − a² + 5 − 2a − 1 = 3a² + a + 4 — note that a² terms, a terms and constants are all different types and cannot be mixed.
Substitution
Substitution means replacing a letter (variable) with a given number. Always use brackets around negative values when substituting — especially before applying a power — to avoid sign errors.
For example: if a = 3 and b = −2, find 4a − 3b²: substitute to get 4(3) − 3(−2)² = 12 − 3(4) = 0.
Full worked examples & practice →Expanding brackets
Expanding means multiplying everything inside a bracket by the term outside. For double brackets, use FOIL (First, Outer, Inner, Last) — every term in the first bracket multiplies every term in the second.
For example: (2x + 3)(x − 4) = 2x² − 8x + 3x − 12 = 2x² − 5x − 12.
Full worked examples & practice →Factorising
Factorising is the reverse of expanding. Always look for the highest common factor (HCF) first. For quadratics ax² + bx + c, find two numbers that multiply to ac and add to b.
For example: factorise 2x² + 5x − 3. Here ac = −6; two numbers that multiply to −6 and add to +5 are +6 and −1. Split: 2x² + 6x − x − 3 = (2x − 1)(x + 3).
Full worked examples & practice →Changing the subject of a formula
To change the subject, use inverse operations to isolate the required variable — exactly as you would when solving an equation. Work systematically: deal with addition/subtraction first, then multiplication/division, then powers/roots.
For example: make x the subject of y = 3x² − 5. Add 5: y + 5 = 3x². Divide by 3: (y + 5)/3 = x². Square root: x = √((y + 5)/3).
Full worked examples & practice →Want to improve your grade faster?
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