Sec 2 G2 Maths Algebraic expressions and formulae worksheet with answers

A free Sec 2 G2 Maths worksheet on algebraic expressions and formulae: 8 questions, 30 marks, short questions first and then structured ones, in the school exam style. The answers and full worked solutions are on the last page.

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Sec 2 G2 Maths worksheet: Algebraic expressions and formulae

Total: 30 marksSuggested time: 50 minutes

Name: ______________________________Class: ____________ Date: ____________

Answer all questions. Show all your working. Give non-exact answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless the question says otherwise.

Section A

(a)Expand and simplify

(x + 2)(3x^{2} + 4x - 5)

Answer (a)[1]

(b)Expand and simplify

(5a - 3b)^{2}

Answer (b)[1]

(a)Simplify

\dfrac{x^{2} + 6x + 5}{x + 1}

Answer (a)[1]

(b)Simplify

\dfrac{x^{2} - 5x - 6}{x + 1}

Answer (b)[1]

(a)Expand and simplify

(4x - 7)(3x - 2)

Answer (a)[1]

(b)Expand and simplify

(4x - 3)(x + 2)

Answer (b)[1]

(a)Factorise

x^{2} - 6x - 16

Answer (a)[1]

(b)Factorise

10x^{2} + 27x - 9

Answer (b)[1]

(a)Factorise

6hm + 8h + 15km + 20k

completely.

Answer (a)[1]

(b)Factorise

16x^{3} - 36x

completely.

Answer (b)[1]

Section B

(a)(i)Expand

(x - 6)^2

Answer (a)(i)[1]

(a)(ii)Expand and simplify

(5x + 1)(5x - 1)

Answer (a)(ii)[1]

(b)(i)Factorise

80x^{2} - 245y^{2}

completely.

Answer (b)(i)[1]

(b)(ii)Factorise

2p^{2} - 5pq + 2q^{2}

completely.

Answer (b)(ii)[2]

(a)Show that

(n + 6)^{2} - (n - 4)^{2}

is a multiple of

20

for every positive integer

n

Answer (a)[2]

(b)(i)Expand and simplify

(5x + 5)(4x + 2)

Answer (b)(i)[1]

(b)(ii)Expand and simplify

(x + 1)(x - 1)

Answer (b)(ii)[1]

(c)(i)Simplify

\dfrac{x^{2} - x - 12}{x + 3}

Answer (c)(i)[1]

(c)(ii)Simplify

\dfrac{x^{2} - 2x - 8}{x^{2} + 2x - 8} \times \dfrac{x + 4}{x + 2}

Answer (c)(ii)[2]

(a)(i)Factorise

x^{2} + 9x + 14

Answer (a)(i)[1]

(a)(ii)Factorise

x^{2} - 7x - 8

Answer (a)(ii)[1]

(b)(i)Factorise

p^{2} - 4q^{2}

completely.

Answer (b)(i)[1]

(b)(ii)Factorise

9x^{2} + 39xy - 30y^{2}

completely.

Answer (b)(ii)[1]

(c)(i)Expand and simplify

(x + 3)(x + 8)

Answer (c)(i)[1]

(c)(ii)Expand and simplify

(x + 4)(x - 4)

Answer (c)(ii)[1]

(d)Express

\dfrac{2}{x - 3} + \dfrac{2}{x + 1}

as a single fraction in its simplest form.

Answer (d)[2]

END OF PAPER

Answers and worked solutions

Question 1

(a)Answer:

= 3x^{3} + 10x^{2} + 3x - 10

[1]

(x + 2)(3x^{2} + 4x - 5) = 3x^{3} + 4x^{2} - 5x + 6x^{2} + 8x - 10

= 3x^{3} + 10x^{2} + 3x - 10

(b)Answer:

(5a - 3b)^{2} = (25a^{2} - 30ab + 9b^{2})

[1]

(5a - 3b)^{2} = (25a^{2} - 30ab + 9b^{2})

Question 2

(a)Answer:

= x + 5

[1]

\dfrac{x^{2} + 6x + 5}{x + 1} = \dfrac{(x + 1)(x + 5)}{(x + 1)}

= x + 5

(b)Answer:

= x - 6

[1]

\dfrac{x^{2} - 5x - 6}{x + 1} = \dfrac{(x + 1)(x - 6)}{(x + 1)}

= x - 6

Question 3

(a)Answer:

12x^2-29x+14

[1]

12x^2-29x+14

(b)Answer:

4x^2+5x-6

[1]

4x^2+5x-6

Question 4

(a)Answer:

(x-8)(x+2)

[1]

(x-8)(x+2)

(b)Answer:

(10x-3)(x+3)

[1]

(10x-3)(x+3)

Question 5

(a)Answer:

= (3m + 4)(2h + 5k)

[1]

6hm + 8h + 15km + 20k = 2h(3m + 4) + 5k(3m + 4)

= (3m + 4)(2h + 5k)

(b)Answer:

= 4x(2x + 3)(2x - 3)

[1]

16x^{3} - 36x = 4x(4x^{2} - 9)

= 4x(2x + 3)(2x - 3)

Question 6

(a)(i)Answer:

x^2-12x+36

[1]

x^2-12x+36

(a)(ii)Answer:

25x^2-1

[1]

25x^2-1

(b)(i)Answer:

= 5(4x + 7y)(4x - 7y)

[1]

80x^{2} - 245y^{2} = 5(16x^{2} - 49y^{2})

= 5(4x + 7y)(4x - 7y)

(b)(ii)Answer:

= (p - 2q)(2p - q)

[2]

-1 \times (-4) = 4

and

-1 + (-4) = -5

2p^{2} - 5pq + 2q^{2} = 2p^{2} - pq - 4pq + 2q^{2}

= p(2p - q) - 2q(2p - q)

= (p - 2q)(2p - q)

Question 7

(a)Answer:

n + 1

is a whole number, so this is a multiple of

20

for every positive integer

n

.[2]

(n + 6)^{2} - (n - 4)^{2} = (n^{2} + 12n + 36) - (n^{2} - 8n + 16)

= 20n + 20

= 20(n + 1)

n + 1

is a whole number, so this is a multiple of

20

for every positive integer

n

(b)(i)Answer:

20x^2+30x+10

[1]

20x^2+30x+10

(b)(ii)Answer:

x^2-1

[1]

x^2-1

(c)(i)Answer:

= x - 4

[1]

\dfrac{x^{2} - x - 12}{x + 3} = \dfrac{(x + 3)(x - 4)}{(x + 3)}

= x - 4

(c)(ii)Answer:

= \dfrac{x - 4}{x - 2}

[2]

\dfrac{x^{2} - 2x - 8}{x^{2} + 2x - 8} \times \dfrac{x + 4}{x + 2}

= \dfrac{x^{2} - 2x - 8}{x^{2} + 2x - 8} \times \dfrac{x + 4}{x + 2}

= \dfrac{(x + 2)(x - 4)}{(x + 4)(x - 2)} \times \dfrac{x + 4}{x + 2}

= \dfrac{x - 4}{x - 2}

Question 8

(a)(i)Answer:

(x+2)(x+7)

[1]

(x+2)(x+7)

(a)(ii)Answer:

(x-8)(x+1)

[1]

(x-8)(x+1)

(b)(i)Answer:

= (p + 2q)(p - 2q)

[1]

p^{2} - 4q^{2} = (p)^{2} - (2q)^{2}

= (p + 2q)(p - 2q)

(b)(ii)Answer:

= 3(x + 5y)(3x - 2y)

[1]

9x^{2} + 39xy - 30y^{2} = 3(3x^{2} + 13xy - 10y^{2})

= 3(x + 5y)(3x - 2y)

(c)(i)Answer:

x^2+11x+24

[1]

x^2+11x+24

(c)(ii)Answer:

x^2-16

[1]

x^2-16

(d)Answer:

= \dfrac{4x - 4}{(x - 3)(x + 1)}

[2]

\dfrac{2}{x - 3} + \dfrac{2}{x + 1} = \dfrac{2(x + 1) + 2(x - 3)}{(x - 3)(x + 1)}

= \dfrac{2x + 2 + (2x - 6)}{(x - 3)(x + 1)}

= \dfrac{4x - 4}{(x - 3)(x + 1)}

Sec 2 G2 Maths Algebraic expressions and formulae worksheet with answers

In the print window, choose Save as PDF to download it.

Or practise it online with step-by-step help when you get one wrong.

Section B

(a)(i)Expand

(x - 6)^2

Answer (a)(i)[1]

(a)(ii)Expand and simplify

(5x + 1)(5x - 1)

Answer (a)(ii)[1]

(b)(i)Factorise

80x^{2} - 245y^{2}

completely.

Answer (b)(i)[1]

(b)(ii)Factorise

2p^{2} - 5pq + 2q^{2}

completely.

Answer (b)(ii)[2]

Answers and worked solutions

Question 1

(a)Answer:

= 3x^{3} + 10x^{2} + 3x - 10

[1]

(x + 2)(3x^{2} + 4x - 5) = 3x^{3} + 4x^{2} - 5x + 6x^{2} + 8x - 10

= 3x^{3} + 10x^{2} + 3x - 10

(b)Answer:

(5a - 3b)^{2} = (25a^{2} - 30ab + 9b^{2})

[1]

(5a - 3b)^{2} = (25a^{2} - 30ab + 9b^{2})

Question 2

(a)Answer:

= x + 5

[1]

\dfrac{x^{2} + 6x + 5}{x + 1} = \dfrac{(x + 1)(x + 5)}{(x + 1)}

= x + 5

(b)Answer:

= x - 6

[1]

\dfrac{x^{2} - 5x - 6}{x + 1} = \dfrac{(x + 1)(x - 6)}{(x + 1)}

= x - 6

Question 3

(a)Answer:

12x^2-29x+14

[1]

12x^2-29x+14

(b)Answer:

4x^2+5x-6

[1]

4x^2+5x-6

Question 4

(a)Answer:

(x-8)(x+2)

[1]

(x-8)(x+2)

(b)Answer:

(10x-3)(x+3)

[1]

(10x-3)(x+3)

Question 5

(a)Answer:

= (3m + 4)(2h + 5k)

[1]

6hm + 8h + 15km + 20k = 2h(3m + 4) + 5k(3m + 4)

= (3m + 4)(2h + 5k)

(b)Answer:

= 4x(2x + 3)(2x - 3)

[1]

16x^{3} - 36x = 4x(4x^{2} - 9)

= 4x(2x + 3)(2x - 3)

Question 6

(a)(i)Answer:

x^2-12x+36

[1]

x^2-12x+36

(a)(ii)Answer:

25x^2-1

[1]

25x^2-1

(b)(i)Answer:

= 5(4x + 7y)(4x - 7y)

[1]

80x^{2} - 245y^{2} = 5(16x^{2} - 49y^{2})

= 5(4x + 7y)(4x - 7y)

(b)(ii)Answer:

= (p - 2q)(2p - q)

[2]

-1 \times (-4) = 4

and

-1 + (-4) = -5

2p^{2} - 5pq + 2q^{2} = 2p^{2} - pq - 4pq + 2q^{2}

= p(2p - q) - 2q(2p - q)

= (p - 2q)(2p - q)

Question 7

(a)Answer:

n + 1

is a whole number, so this is a multiple of

20

for every positive integer

n

.[2]

(n + 6)^{2} - (n - 4)^{2} = (n^{2} + 12n + 36) - (n^{2} - 8n + 16)

= 20n + 20

= 20(n + 1)

n + 1

is a whole number, so this is a multiple of

20

for every positive integer

n

(b)(i)Answer:

20x^2+30x+10

[1]

20x^2+30x+10

(b)(ii)Answer:

x^2-1

[1]

x^2-1

(c)(i)Answer:

= x - 4

[1]

\dfrac{x^{2} - x - 12}{x + 3} = \dfrac{(x + 3)(x - 4)}{(x + 3)}

= x - 4

(c)(ii)Answer:

= \dfrac{x - 4}{x - 2}

[2]

\dfrac{x^{2} - 2x - 8}{x^{2} + 2x - 8} \times \dfrac{x + 4}{x + 2}

= \dfrac{x^{2} - 2x - 8}{x^{2} + 2x - 8} \times \dfrac{x + 4}{x + 2}

= \dfrac{(x + 2)(x - 4)}{(x + 4)(x - 2)} \times \dfrac{x + 4}{x + 2}

= \dfrac{x - 4}{x - 2}

Question 8

(a)(i)Answer:

(x+2)(x+7)

[1]

(x+2)(x+7)

(a)(ii)Answer:

(x-8)(x+1)

[1]

(x-8)(x+1)

(b)(i)Answer:

= (p + 2q)(p - 2q)

[1]

p^{2} - 4q^{2} = (p)^{2} - (2q)^{2}

= (p + 2q)(p - 2q)

(b)(ii)Answer:

= 3(x + 5y)(3x - 2y)

[1]

9x^{2} + 39xy - 30y^{2} = 3(3x^{2} + 13xy - 10y^{2})

= 3(x + 5y)(3x - 2y)

(c)(i)Answer:

x^2+11x+24

[1]

x^2+11x+24

(c)(ii)Answer:

x^2-16

[1]

x^2-16

(d)Answer:

= \dfrac{4x - 4}{(x - 3)(x + 1)}

[2]

\dfrac{2}{x - 3} + \dfrac{2}{x + 1} = \dfrac{2(x + 1) + 2(x - 3)}{(x - 3)(x + 1)}

= \dfrac{2x + 2 + (2x - 6)}{(x - 3)(x + 1)}

= \dfrac{4x - 4}{(x - 3)(x + 1)}