GCSE Maths Algebraic manipulation questions by topic

Simplifying expressions.

1. (a)Simplify 7p + 4q − 3p + q.[1]
2. (b)Simplify 3x × 4x².[1]
3. (c)Expand 6(2y − 5).[2]

Perimeters with algebra.

1. (a)A rectangle has sides of length (2x + 3) cm and (x + 1) cm. Write an expression for its perimeter.[2]
2. (b)Expand 4(3x − 2).[2]

Expanding and spotting vocabulary.

1. (a)Expand −2(5 − 3x).[2]
2. (b)Expand and simplify 2(3x + 1) − 3(x − 4).[3]

Factorising with common factors.

1. (a)Factorise 5x + 20.[1]
2. (b)Factorise 8x² − 12x.[2]
3. (c)Factorise fully 3a²b + 6ab².[1]

A rectangle has algebraic sides.

1. (a)The rectangle is 3x cm long and (2x + 1) cm wide. Write down and simplify an expression for its perimeter.[2]
2. (b)Write down and simplify an expression for its area.[2]

A quadratic function is evaluated at two points.

1. (a)For y = 2x² − 5x + 1, work out y when x = 3.[2]
2. (b)Substitute x = −1 into the same function to find y.[2]

Double brackets.

1. (a)Expand (x + 5)(x − 2).[2]
2. (b)Expand and simplify (x + 3)² + (x + 3)(x − 3).[3]

Quadratic factorising.

1. (a)Factorise x² + 8x + 15.[2]
2. (b)Factorise x² − 3x − 28.[2]
3. (c)Factorise x² − 100.[1]

An area puzzle with algebra.

1. (a)A square has side (x + 2) cm. Write an expanded expression for its area.[2]
2. (b)A rectangle has the same area as the square but sides x and (x + k). Find k.[2]

Expanding three factors, step by step.

1. (a)Expand and simplify (x + 1)(x + 2)(x − 3).[4]

Factorise, then check by expanding.

1. (a)Factorise x² − 3x − 28.[2]
2. (b)Expand your answer to show that it gives x² − 3x − 28.[2]

Brackets are expanded and compared.

1. (a)Multiply out (2x + 3)(x − 4) and collect like terms.[2]
2. (b)Expand the difference-of-two-squares brackets (x − 5)(x + 5).[1]
3. (c)Explain why the expansion in part (b) has no x term.[2]

Factorising ax² + bx + c. (Higher)

1. (a)Factorise 2x² + 7x + 3.[2]
2. (b)Factorise 3x² − 5x − 2.[2]
3. (c)Factorise fully 6x³ − 24x.[2]

Algebraic fractions. (Higher)

1. (a)Simplify (x² + 5x + 6) ÷ (x² − 4).[3]
2. (b)Write 2⁄(x − 1) + 3⁄(x + 2) as a single fraction in its simplest form.[3]

Indices in algebra. (Higher)

1. (a)Simplify (4x³y²)² × 3x⁻⁴y.[3]
2. (b)Simplify 18x⁷ ÷ 6x³, giving your answer with a positive index.[1]
3. (c)Simplify (16x⁸)^(1/4).[2]

A surd expansion proof. (Higher)

1. (a)Expand and simplify (√3 + 1)(√3 − 2).[3]
2. (b)Show that (√7 + √2)² + (√7 − √2)² = 18.[4]

This question is about harder quadratics. (Higher)

1. (a)Factorise 6x² + 19x + 15.[2]
2. (b)Factorise 9x² − 12x + 4.[2]
3. (c)Hence solve 9x² − 12x + 4 = 0.[2]

Fractions and surds in one question. (Higher)

1. (a)Write 5⁄(x + 2) + 3⁄(x − 1) as a single fraction, simplifying your answer.[3]
2. (b)Multiply out and simplify the surd product (√7 + 1)(√7 − 4).[3]