Sec 3 G2 Maths Algebraic manipulation worksheet with answers

A free Sec 3 G2 Maths worksheet on algebraic manipulation: 10 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 3 G2 Maths worksheet: Algebraic manipulation

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)Factorise

x^{2} - 3x - 54

Answer (a)[1]

(b)Factorise

3x^{2} - 2x - 1

Answer (b)[1]

Simplify

\dfrac{x^{2} + 5x + 6}{x^{2} + 2x - 8} \times \dfrac{x + 4}{x + 2}

Answer[2]

(a)Expand and simplify

(3x + 8)(3x - 8)

Answer (a)[1]

(b)Expand and simplify

(2x + 2)(2x - 2)

Answer (b)[1]

(a)Factorise

15xy - 20x - 3y + 4

completely.

Answer (a)[1]

(b)Given that

3x^{2} - 20x + 25 = (3x - 5)(x - 5)

, factorise

3(y - 3)^{2} - 20(y - 3) + 25

completely.

Answer (b)[1]

(a)Make

x

the subject of

y = -5x + 1

Answer (a)[1]

(b)Make

x

the subject of

y = 4\sqrt{x} - 4

Answer (b)[2]

Section B

Make

x

the subject of the formula

y = \dfrac{3x + 8}{x + 5}

Answer[3]

Make

x

the subject of the formula

y = \dfrac{5x - 4}{x - 3}

Answer[3]

(a)Expand and simplify

(3x - 7)(2x + 3) - 3x(x - 2)

Answer (a)[2]

(b)Factorise completely

3x^{2} - 3

Answer (b)[2]

(a)Expand and simplify

(x + 2)(2x - 7) - 3x(x + 2)

Answer (a)[2]

(b)Factorise completely

5x^{2} - 80

Answer (b)[2]

(a)Simplify

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

Answer (a)[2]

(b)Express

\dfrac{2}{x + 5} + \dfrac{4}{x - 1}

as a single fraction in its simplest form.

Answer (b)[3]

END OF PAPER

Answers and worked solutions

Question 1

(a)Answer:

(x-9)(x+6)

[1]

(x-9)(x+6)

(b)Answer:

(3x+1)(x-1)

[1]

(3x+1)(x-1)

Question 2

Answer:

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

[2]

\dfrac{x^{2} + 5x + 6}{x^{2} + 2x - 8} \times \dfrac{x + 4}{x + 2}

= \dfrac{x^{2} + 5x + 6}{x^{2} + 2x - 8} \times \dfrac{x + 4}{x + 2}

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

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

Question 3

(a)Answer:

9x^2-64

[1]

9x^2-64

(b)Answer:

4x^2-4

[1]

4x^2-4

Question 4

(a)Answer:

= (3y - 4)(5x - 1)

[1]

15xy - 20x - 3y + 4 = 5x(3y - 4) - 1(3y - 4)

= (3y - 4)(5x - 1)

(b)Answer:

= (3y - 14)(y - 8)

[1]

3(y - 3)^{2} - 20(y - 3) + 25 = [3(y - 3) - 5][(y - 3) - 5]

= (3y - 14)(y - 8)

Question 5

(a)Answer:

x = \dfrac{y - 1}{-5}

[1]

y - 1 = -5x

x = \dfrac{y - 1}{-5}

(b)Answer:

x = \left(\dfrac{y + 4}{4}\right)^{2}

[2]

y + 4 = 4\sqrt{x}

\sqrt{x} = \dfrac{y + 4}{4}

x = \left(\dfrac{y + 4}{4}\right)^{2}

Question 6

Answer:

x = \dfrac{8 - 5y}{y - 3}

[3]

y(x + 5) = 3x + 8

xy + 5y = 3x + 8

xy - 3x = 8 - 5y

x(y - 3) = 8 - 5y

x = \dfrac{8 - 5y}{y - 3}

Question 7

Answer:

x = \dfrac{-4 + 3y}{y - 5}

[3]

y(x - 3) = 5x - 4

xy - 3y = 5x - 4

xy - 5x = -4 + 3y

x(y - 5) = -4 + 3y

x = \dfrac{-4 + 3y}{y - 5}

Question 8

(a)Answer:

3x^{2} + x - 21

[2]

(3x - 7)(2x + 3) - 3x(x - 2)

= (6x^{2} - 5x - 21) - (3x^{2} - 6x)

= 6x^{2} -5x -21 -3x^{2} +6x

= 3x^{2} + x - 21

(b)Answer:

3(x + 1)(x - 1)

[2]

3x^{2} - 3

= 3(x^{2} - 1)

(take out the common factor first)

= 3(x + 1)(x - 1)

(difference of two squares:

a^2 - b^2 = (a + b)(a - b)

)

Question 9

(a)Answer:

-x^{2} - 9x - 14

[2]

(x + 2)(2x - 7) - 3x(x + 2)

= (2x^{2} - 3x - 14) - (3x^{2} + 6x)

= 2x^{2} -3x -14 -3x^{2} -6x

= -x^{2} - 9x - 14

(b)Answer:

5(x + 4)(x - 4)

[2]

5x^{2} - 80

= 5(x^{2} - 16)

(take out the common factor first)

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

(difference of two squares:

a^2 - b^2 = (a + b)(a - b)

)

Question 10

(a)Answer:

\dfrac{x - 5}{x + 6}

[2]

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

= \dfrac{x - 5}{x + 6}

(cancel the common bracket

(x - 1)

)

(b)Answer:

\dfrac{6x + 18}{(x + 5)(x - 1)}

[3]

\dfrac{2}{x + 5} + \dfrac{4}{x - 1} = \dfrac{2(x - 1) + 4(x + 5)}{(x + 5)(x - 1)}

= \dfrac{2x -2 +4x +20}{(x + 5)(x - 1)}

= \dfrac{6x + 18}{(x + 5)(x - 1)}

Sec 3 G2 Maths Algebraic manipulation worksheet with answers

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

Need the formulas first? Read the Algebraic manipulation notes. Or practise it online with step-by-step help when you get one wrong.

Answers and worked solutions

Question 1

(a)Answer:

(x-9)(x+6)

[1]

(x-9)(x+6)

(b)Answer:

(3x+1)(x-1)

[1]

(3x+1)(x-1)

Question 2

Answer:

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

[2]

\dfrac{x^{2} + 5x + 6}{x^{2} + 2x - 8} \times \dfrac{x + 4}{x + 2}

= \dfrac{x^{2} + 5x + 6}{x^{2} + 2x - 8} \times \dfrac{x + 4}{x + 2}

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

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

Question 3

(a)Answer:

9x^2-64

[1]

9x^2-64

(b)Answer:

4x^2-4

[1]

4x^2-4

Question 4

(a)Answer:

= (3y - 4)(5x - 1)

[1]

15xy - 20x - 3y + 4 = 5x(3y - 4) - 1(3y - 4)

= (3y - 4)(5x - 1)

(b)Answer:

= (3y - 14)(y - 8)

[1]

3(y - 3)^{2} - 20(y - 3) + 25 = [3(y - 3) - 5][(y - 3) - 5]

= (3y - 14)(y - 8)

Question 5

(a)Answer:

x = \dfrac{y - 1}{-5}

[1]

y - 1 = -5x

x = \dfrac{y - 1}{-5}

(b)Answer:

x = \left(\dfrac{y + 4}{4}\right)^{2}

[2]

y + 4 = 4\sqrt{x}

\sqrt{x} = \dfrac{y + 4}{4}

x = \left(\dfrac{y + 4}{4}\right)^{2}

Question 6

Answer:

x = \dfrac{8 - 5y}{y - 3}

[3]

y(x + 5) = 3x + 8

xy + 5y = 3x + 8

xy - 3x = 8 - 5y

x(y - 3) = 8 - 5y

x = \dfrac{8 - 5y}{y - 3}

Question 7

Answer:

x = \dfrac{-4 + 3y}{y - 5}

[3]

y(x - 3) = 5x - 4

xy - 3y = 5x - 4

xy - 5x = -4 + 3y

x(y - 5) = -4 + 3y

x = \dfrac{-4 + 3y}{y - 5}

Question 8

(a)Answer:

3x^{2} + x - 21

[2]

(3x - 7)(2x + 3) - 3x(x - 2)

= (6x^{2} - 5x - 21) - (3x^{2} - 6x)

= 6x^{2} -5x -21 -3x^{2} +6x

= 3x^{2} + x - 21

(b)Answer:

3(x + 1)(x - 1)

[2]

3x^{2} - 3

= 3(x^{2} - 1)

(take out the common factor first)

= 3(x + 1)(x - 1)

(difference of two squares:

a^2 - b^2 = (a + b)(a - b)

)

Question 9

(a)Answer:

-x^{2} - 9x - 14

[2]

(x + 2)(2x - 7) - 3x(x + 2)

= (2x^{2} - 3x - 14) - (3x^{2} + 6x)

= 2x^{2} -3x -14 -3x^{2} -6x

= -x^{2} - 9x - 14

(b)Answer:

5(x + 4)(x - 4)

[2]

5x^{2} - 80

= 5(x^{2} - 16)

(take out the common factor first)

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

(difference of two squares:

a^2 - b^2 = (a + b)(a - b)

)

Question 10

(a)Answer:

\dfrac{x - 5}{x + 6}

[2]

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

= \dfrac{x - 5}{x + 6}

(cancel the common bracket

(x - 1)

)

(b)Answer:

\dfrac{6x + 18}{(x + 5)(x - 1)}

[3]

\dfrac{2}{x + 5} + \dfrac{4}{x - 1} = \dfrac{2(x - 1) + 4(x + 5)}{(x + 5)(x - 1)}

= \dfrac{2x -2 +4x +20}{(x + 5)(x - 1)}

= \dfrac{6x + 18}{(x + 5)(x - 1)}