Sec 3 E-Math Algebraic manipulation worksheet with answers

A free Sec 3 E-Math 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 E-Math worksheet: Algebraic manipulation

Total: 30 marksSuggested time: 45 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

64m^{2} - 25n^{2}

completely.

Answer (a)[1]

(b)Factorise

x^{2} - 9y^{2} - 5x - 15y

completely.

Answer (b)[1]

(a)Expand and simplify

(x + 5)(x + 7)

Answer (a)[1]

(b)Expand and simplify

(5x - 1)(x - 1)

Answer (b)[1]

(a)Factorise

x^{2} - x - 56

Answer (a)[1]

(b)Factorise

7x^{2} - 23x - 20

Answer (b)[1]

Show that

(3n + 6)^{2} - (3n - 6)^{2}

is a multiple of

72

for every positive integer

n

Answer[2]

(a)Make

x

the subject of

y = -2x - 1

Answer (a)[1]

(b)Make

x

the subject of

y = 3\sqrt{x} + 5

Answer (b)[2]

Section B

Make

x

the subject of the formula

y = \dfrac{5x + 1}{x - 2}

Answer[3]

Make

x

the subject of the formula

y = \dfrac{3x + 1}{x - 3}

Answer[3]

(a)Expand and simplify

(2x - 3)(2x - 3) - 3x(x + 6)

Answer (a)[2]

(b)Factorise completely

6x^{2} + 9x - 6

Answer (b)[2]

(a)Expand and simplify

(3x - 1)^2 - 2(x + 1)(x - 2)

Answer (a)[2]

(b)Factorise completely

6x^{2} - 26x - 60

Answer (b)[2]

(a)Simplify

\dfrac{x^{2} + 11x + 30}{x^{2} + 3x - 18}

Answer (a)[2]

(b)Express

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

as a single fraction in its simplest form.

Answer (b)[3]

END OF PAPER

Answers and worked solutions

Question 1

(a)Answer:

= (8m + 5n)(8m - 5n)

[1]

64m^{2} - 25n^{2} = (8m)^{2} - (5n)^{2}

= (8m + 5n)(8m - 5n)

(b)Answer:

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

[1]

x^{2} - 9y^{2} - 5x - 15y = (x + 3y)(x - 3y) - 5(x + 3y)

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

Question 2

(a)Answer:

x^2+12x+35

[1]

x^2+12x+35

(b)Answer:

5x^2-6x+1

[1]

5x^2-6x+1

Question 3

(a)Answer:

(x-8)(x+7)

[1]

(x-8)(x+7)

(b)Answer:

(7x+5)(x-4)

[1]

(7x+5)(x-4)

Question 4

Answer:

= 72 \times n

72

times a whole number, so it is a multiple of

72

for every positive integer

n

.[2]

(3n + 6)^{2} - (3n - 6)^{2} = (9n^{2} + 36n + 36) - (9n^{2} - 36n + 36)

= 72n

= 72 \times n

72

times a whole number, so it is a multiple of

72

for every positive integer

n

Question 5

(a)Answer:

x = \dfrac{y + 1}{-2}

[1]

y + 1 = -2x

x = \dfrac{y + 1}{-2}

(b)Answer:

x = \left(\dfrac{y - 5}{3}\right)^{2}

[2]

y - 5 = 3\sqrt{x}

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

x = \left(\dfrac{y - 5}{3}\right)^{2}

Question 6

Answer:

x = \dfrac{1 + 2y}{y - 5}

[3]

y(x - 2) = 5x + 1

xy - 2y = 5x + 1

xy - 5x = 1 + 2y

x(y - 5) = 1 + 2y

x = \dfrac{1 + 2y}{y - 5}

Question 7

Answer:

x = \dfrac{1 + 3y}{y - 3}

[3]

y(x - 3) = 3x + 1

xy - 3y = 3x + 1

xy - 3x = 1 + 3y

x(y - 3) = 1 + 3y

x = \dfrac{1 + 3y}{y - 3}

Question 8

(a)Answer:

x^{2} - 30x + 9

[2]

(2x - 3)(2x - 3) - 3x(x + 6)

= (4x^{2} - 12x + 9) - (3x^{2} + 18x)

= 4x^{2} -12x +9 -3x^{2} -18x

= x^{2} - 30x + 9

(b)Answer:

3(2x - 1)(x + 2)

[2]

6x^{2} + 9x - 6

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

(take out the common factor first)

= 3(2x - 1)(x + 2)

Question 9

(a)Answer:

7x^{2} - 4x + 5

[2]

(3x - 1)^2 - 2(x + 1)(x - 2)

= (9x^{2} - 6x + 1) - (2x^{2} - 2x - 4)

= 9x^{2} -6x +1 -2x^{2} +2x +4

= 7x^{2} - 4x + 5

(b)Answer:

2(3x + 5)(x - 6)

[2]

6x^{2} - 26x - 60

= 2(3x^{2} - 13x - 30)

(take out the common factor first)

= 2(3x + 5)(x - 6)

Question 10

(a)Answer:

\dfrac{x + 5}{x - 3}

[2]

\dfrac{x^{2} + 11x + 30}{x^{2} + 3x - 18} = \dfrac{(x + 6)(x + 5)}{(x + 6)(x - 3)}

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

(cancel the common bracket

(x + 6)

)

(b)Answer:

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

[3]

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

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

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

Practice questions by skill

Sec 3 E-Math 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:

= (8m + 5n)(8m - 5n)

[1]

64m^{2} - 25n^{2} = (8m)^{2} - (5n)^{2}

= (8m + 5n)(8m - 5n)

(b)Answer:

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

[1]

x^{2} - 9y^{2} - 5x - 15y = (x + 3y)(x - 3y) - 5(x + 3y)

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

Question 2

(a)Answer:

x^2+12x+35

[1]

x^2+12x+35

(b)Answer:

5x^2-6x+1

[1]

5x^2-6x+1

Question 3

(a)Answer:

(x-8)(x+7)

[1]

(x-8)(x+7)

(b)Answer:

(7x+5)(x-4)

[1]

(7x+5)(x-4)

Question 4

Answer:

= 72 \times n

72

times a whole number, so it is a multiple of

72

for every positive integer

n

.[2]

(3n + 6)^{2} - (3n - 6)^{2} = (9n^{2} + 36n + 36) - (9n^{2} - 36n + 36)

= 72n

= 72 \times n

72

times a whole number, so it is a multiple of

72

for every positive integer

n

Question 5

(a)Answer:

x = \dfrac{y + 1}{-2}

[1]

y + 1 = -2x

x = \dfrac{y + 1}{-2}

(b)Answer:

x = \left(\dfrac{y - 5}{3}\right)^{2}

[2]

y - 5 = 3\sqrt{x}

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

x = \left(\dfrac{y - 5}{3}\right)^{2}

Question 6

Answer:

x = \dfrac{1 + 2y}{y - 5}

[3]

y(x - 2) = 5x + 1

xy - 2y = 5x + 1

xy - 5x = 1 + 2y

x(y - 5) = 1 + 2y

x = \dfrac{1 + 2y}{y - 5}

Question 7

Answer:

x = \dfrac{1 + 3y}{y - 3}

[3]

y(x - 3) = 3x + 1

xy - 3y = 3x + 1

xy - 3x = 1 + 3y

x(y - 3) = 1 + 3y

x = \dfrac{1 + 3y}{y - 3}

Question 8

(a)Answer:

x^{2} - 30x + 9

[2]

(2x - 3)(2x - 3) - 3x(x + 6)

= (4x^{2} - 12x + 9) - (3x^{2} + 18x)

= 4x^{2} -12x +9 -3x^{2} -18x

= x^{2} - 30x + 9

(b)Answer:

3(2x - 1)(x + 2)

[2]

6x^{2} + 9x - 6

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

(take out the common factor first)

= 3(2x - 1)(x + 2)

Question 9

(a)Answer:

7x^{2} - 4x + 5

[2]

(3x - 1)^2 - 2(x + 1)(x - 2)

= (9x^{2} - 6x + 1) - (2x^{2} - 2x - 4)

= 9x^{2} -6x +1 -2x^{2} +2x +4

= 7x^{2} - 4x + 5

(b)Answer:

2(3x + 5)(x - 6)

[2]

6x^{2} - 26x - 60

= 2(3x^{2} - 13x - 30)

(take out the common factor first)

= 2(3x + 5)(x - 6)

Question 10

(a)Answer:

\dfrac{x + 5}{x - 3}

[2]

\dfrac{x^{2} + 11x + 30}{x^{2} + 3x - 18} = \dfrac{(x + 6)(x + 5)}{(x + 6)(x - 3)}

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

(cancel the common bracket

(x + 6)

)

(b)Answer:

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

[3]

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

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

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