Edexcel IGCSE Math A Higher 2.2 Algebraic Manipulation Questions

Use this manipulation page to collect terms, expand brackets, factorise expressions and rewrite algebra before applying it in equations.

Exam points

Collect like terms and expand brackets before simplifying the final expression.
Factorise common factors and quadratics, including x² + bx + c forms with signs.
Rewrite expressions into a requested form before substituting or solving.

Covered syllabus points

2.2.A Evaluate expressions by substituting numerical values 2.2.B Collect like terms 2.2.C Expanding a single term over a bracket 2.2.D Taking out common factors 2.2.E Products of two linear expressions 2.2.F Factorising simple quadratic expressions 2.2.HA Products of multiple linear expressions 2.2.HB Factorising quadratic expressions 2.2.HC Algebraic fractions 2.2.HD Completing the square 2.2.HE Algebra to support and construct proofs

[Maximum number: 1]

Given that $8 \sqrt{m}+\sqrt{49 m}-\sqrt{9 m}=k \sqrt{m}$ where k is an integer and m is a prime number,

work out the value of k

[Maximum number: 6]

(a)

Factorise fully $8 p^{2}-2 p$

[ 2 ]

$8p^2-2p=2p(4p-1)$
Final answer: 2p(4p-1)

(b)

Expand 4 t(3 t-2)

[ 2 ]

$4t(3t-2)=12t^2-8t$
Final answer: $12t^2-8t$

(c)

Expand and simplify (5 x-2)(x+4)

\section*{ $1 \quad w=5 y^{2}-y^{3}$ \\ $1 \quad w=5 y^{2}-y^{3}$ }
(a) Work out the value of w when y=-2

[ 2 ]

$(5x-2)(x+4)=5x^2+20x-2x-8$ $=5x^2+18x-8$
Final answer: $5x^2+18x-8$

[Maximum number: 4]

(a)

Factorise fully $16 m^{3} g^{3}+24 m^{2} g^{5}$

[ 2 ]

$8m^2g^3(2m+3g^2)$

Marking guidance:
- M1 for any correct partial factorisation with at least 2 factors, one of which must be a letter, or the correct common factor with no more than 1 error inside the bracket
- A1 for $8m^2g^3(2m+3g^2)$

(b)

Factorise $y^{2}-2 y-48$

[ 2 ]

(y-8)(y+6)

Marking guidance:
- M1 for $(y\pm6)(y\pm8)$
- A1 for (y-8)(y+6)

[Maximum number: 2]

Expand 4 t(3 t-2)

$4t(3t-2)=12t^2-8t$
Final answer: $12t^2-8t$

[Maximum number: 2]

Factorise fully $16 m^{3} g^{3}+24 m^{2} g^{5}$

[Maximum number: 2]

Expand and simplify (5 x-2)(x+4)

\section*{ $1 \quad w=5 y^{2}-y^{3}$ \\ $1 \quad w=5 y^{2}-y^{3}$ }
(a) Work out the value of w when y=-2

$(5x-2)(x+4)=5x^2+20x-2x-8$ $=5x^2+18x-8$
Final answer: $5x^2+18x-8$

[Maximum number: 2]

Factorise $y^{2}-2 y-48$

(y-8)(y+6)

Marking guidance:
- M1 for $(y\pm6)(y\pm8)$
- A1 for (y-8)(y+6)

[Maximum number: 3]

Expand and simplify 2 x(x-5)(x-3)

$2x(x-5)(x-3)=2x(x^2-8x+15)$ $=2x^3-16x^2+30x$

Final answer:
$2x^3-16x^2+30x$

[Maximum number: 2]

Factorise fully $2 e^{2}-18$

$2e^2-18=2(e^2-9)$ $2(e^2-9)=2(e-3)(e+3)$

Final answer:
2(e-3)(e+3)

[Maximum number: 3]

Express $\frac{4}{x-2}-\frac{3}{x+1}$ as a single fraction.
Give your answer in its simplest form.

$\frac{4}{x-2}-\frac{3}{x+1} =\frac{4(x+1)-3(x-2)}{(x-2)(x+1)}$ $=\frac{4x+4-3x+6}{(x-2)(x+1)} =\frac{x+10}{(x-2)(x+1)}$

Equivalent denominator accepted:
$\frac{x+10}{x^2-x-2}$

[Maximum number: 2]

The graph of $y=\sin x^{\circ}$ for $0 \leqslant x \leqslant 360$ is drawn on the grid.

Figure for Question 11 — Edexcel IGCSE Math A Higher

Write $x^{2}-6 x+10$ in the form $(x-a)^{2}+b$ where a and b are integers.

$(x-3)^2+1$

Marking guidance:
- B2 for $(x-3)^2+1$
- If not B2, award B1 for $\left(x-\frac{6}{2}\right)^2+n$ , where $n\ne1$ , or $(x-m)^2+1$ , where $m\ne3$ , or a valid completing-square expansion with 2a=6 or $a^2+b=10$

[Maximum number: 3]

Here are the first four terms of a sequence of fractions.

\frac{1}{1} \quad \frac{2}{3} \quad \frac{3}{5} \quad \frac{4}{7}

The numerators of the fractions form the sequence of whole numbers $1234 \ldots$ The denominators of the fractions form the sequence of odd numbers $1 \quad 3 \quad 5 \quad 7 \ldots$

Using algebra, prove that when the square of any odd number is divided by 4 the remainder is 1

Let an odd number be 2n-1.
$(2n-1)^2=4n^2-4n+1$ $=4(n^2-n)+1$
Therefore when the square is divided by 4, the remainder is 1.

M1 for using $(2n-1)^2$ or $(2n+1)^2$ .
M1 for writing the expression as a multiple of 4 plus 1.
A1 for a correct conclusion.

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Covered syllabus points

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