OxfordAQA IGCSE Mathematics: Algebra – Notation and Manipulation — Practice Questions (9260)

These are original questions written for Marlbridge, in the style and at the standard of the examination. They are not reproduced past-paper questions — examination boards hold copyright in their own papers. Use these alongside the official past papers available free from your board.

Related: Notation and manipulation study guide

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Section A (Core and Extension)

1. Expand and simplify 3(x + 2) − 2(x − 1). [2]

2. Factorise x² − 9. [2]

Section B (Extension)

3. Make x the subject of the formula: y = (3x + 2)/(x − 1). [4]

4. Factorise 6x² + 5x − 6. [3]

5. Simplify (2x² + 3x)/(4x² − 9). [4]

6. Show that (x + 1)² − (x − 1)² is equivalent to 4x for all values of x. [3]

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Answers

1. 3(x + 2) − 2(x − 1) = 3x + 6 − 2x + 2 = x + 8 [1] [1].

2. x² − 9 = (x − 3)(x + 3) [2] (difference of two squares).

3.

y(x - 1) = 3x + 2                         [1]
xy - y = 3x + 2                           [1]
xy - 3x = y + 2                           [1]
x(y - 3) = y + 2
x = (y + 2) / (y - 3)                     [1]

4. 6x² + 5x − 6 = (2x + 3)(3x − 2) [3] (1 mark for identifying correct factors of 6 and −6 to trial, 1 mark for a partially correct pair of brackets, 1 mark for the fully correct factorisation, verifiable by expanding back out).

5.

Numerator:   2x^2 + 3x = x(2x + 3)                [1]
Denominator: 4x^2 - 9 = (2x - 3)(2x + 3)           [1]
Cancel shared (2x + 3) factor:                     [1]
Result: x / (2x - 3)                               [1]

6.

(x + 1)^2 - (x - 1)^2
= (x^2 + 2x + 1) - (x^2 - 2x + 1)          [1]
= x^2 + 2x + 1 - x^2 + 2x - 1              [1]
= 4x                                        [1]

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Exam technique for this topic

Rearranging formulae questions like Q3, where the subject appears twice, are best approached by first multiplying out to clear any fractions, then collecting all terms containing the target letter on one side of the equation and factoring it out, as shown in the worked answer — attempting to isolate the letter before it has been collected onto one side is a common source of errors on this specific Extension-tier skill. When factorising a quadratic of the form ax² + bx + c where a is not 1 (as in Q4), systematically trial pairs of factors of a and c together rather than guessing, since the extra step compared with factorising x² + bx + c is finding factors of a as well as c — always check your answer by expanding the brackets back out to confirm it matches the original expression. For “show that” questions like Q6, write out every algebraic step explicitly rather than jumping to the answer, since these questions are marked on the working shown, not just on stating that the two expressions are equivalent.

Worked example: a second rearranging-the-subject question

Make t the subject of A = π r² + 2π r t.

A = pi*r^2 + 2*pi*r*t
A - pi*r^2 = 2*pi*r*t              [1]  (isolate the term containing t)
t = (A - pi*r^2) / (2*pi*r)        [1]  (divide both sides by the coefficient of t)

Here t only appears once, so the method is more direct than Q3: isolate the term containing the target letter first by moving every other term to the opposite side, then divide by whatever is multiplying the letter. The harder case, as in Q3, is when the subject appears in two separate terms — that always requires the extra factorising step to bring both occurrences together before a final division can isolate it. Recognising which of these two situations a question presents, before starting to rearrange, saves time and avoids the common error of trying to divide before every occurrence of the target letter has been collected onto one side.

Practising algebraic fraction simplification

Algebraic fractions like Q5 combine two separate skills that must both be correct: factorising the numerator and denominator accurately, and then cancelling only a shared factor (something multiplied), never a shared term (something added or subtracted). A frequent error is to cancel an x that appears in both the numerator and denominator even when it is only part of a larger expression — for example, wrongly cancelling the x in (x + 3)/x² to leave (x + 3)/x, when x is not a factor of the whole numerator (x + 3) and so cannot legally be cancelled, or cancelling terms from 2x² + 3x over 4x² − 9 before those expressions have been fully factorised into their bracketed forms. The safest method is always: factorise fully first, write out both numerator and denominator as products of factors, and only then cross out any factor that appears identically, unchanged, in both.

Where marks are usually lost

• Sign errors when expanding brackets with a negative term, particularly (x − 1)² where the middle term’s sign is easy to get wrong.
• Forgetting that a common factor must be taken out of every term before factorising, not just some of them.
• Applying index laws or cancelling rules to terms with different bases, or cancelling additive terms in a fraction as though they were multiplicative factors.
• In “show that” questions, jumping straight to the final answer without showing the intermediate algebraic steps that the marks are actually awarded for.