IGCSEMaths: ExtendedCIETopic QuestionsAlgebra & SequencesForming & Solving Equations

Syllabus Edition

First teaching 2023

First exams 2025

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Forming & Solving Equations (CIE IGCSE Maths: Extended)

Topic Questions

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1
Sme Calculator3 marks

Solve.

fraction numerator 3 y over denominator 2 y minus 1 end fraction space equals space 3 over 4

y = ..............................................

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2
Sme Calculator10 marks

cie-igcse-spec-p4-q5a-diagram

The perimeter of the rectangle is 80 cm.
The area of the rectangle is Acm2.

i)
Show that  x squared space space – space 40 x space plus space A space equals space 0.

[3]
ii)
When A space equals space 300, solve the equation space x squared space space – space 40 x space plus space A space equals space 0 space by factorising.
  
x = .................. or x = .................  [3]
iii)
When A space equals space 200, solve the equation space space x squared space space – space 40 x space plus space A space equals space 0 space using the quadratic formula.
Show all your working and give your answers correct to 2 decimal places.
  
x= .................. or x = .................  [4]

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3
Sme Calculator9 marks

cie-igcse-2019-oct-nov-p4-tz2-q6b

The area of triangle DEF is 70 m2.

i)
Show that  4 x squared plus 11 x minus 300 equals 0.

 

[4]

ii)
Use the quadratic formula to solve 4 x squared plus 11 x minus 300 equals 0.
Show all your working and give your answers correct to 2 decimal places.

 

x = .......................... or x = .......................... [4]

iii)
Find the length of DE.

 

DE = ............................................... m [1]

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4
Sme Calculator3 marks

A cone with height 14.8 cm has volume 275 cm3.
 
Calculate the radius of the cone.
[The volume, V, of a cone with radius r and height h is V equals 1 third straight pi r squared h.]
 

............................................... cm

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5a
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Paulo and Jim each buy sacks of rice but from different shops.
Paulo pays $72 for sacks costing $m each.
Jim pays $72 for sacks costing $left parenthesis m plus 0.9 right parenthesis each.

i)
Find an expression, in terms of m, for the number of sacks Paulo buys.
 
 

[1]

ii)
Find an expression, in terms of m, for the number of sacks Jim buys.
 
 

[1]

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5b
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Paulo buys 4 more sacks than Jim.
 
Write down an equation, in terms of m, and show that it simplifies to  10 m squared plus 9 m minus 162 equals 0.

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5c
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i)
Solve  10 m squared plus 9 m minus 162 equals 0. 
 
 

m = .................... or m = .................... [3]

ii)
Find the number of sacks of rice that Paulo buys.
 
 

[1]

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6a
Sme Calculator3 marks

The diagram shows a right-angled triangle ABC.

cie-igcse-2018-oct-nov-p4-tz2-q4

The area of this triangle is 30 cm2.

Show that  2 x squared plus 3 x minus 20 equals 0.

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6b
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Use factorisation to solve the equation  2 x squared plus 3 x minus 20 equals 0.
  

x = ...................... or x = ...................... 

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6c
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Calculate BC.
 

BC = .......................................... cm 

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7a
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In this question, all measurements are in metres.

cie-igcse-2018-may-jun-2-p4-q7
The diagram shows a right-angled triangle.

Show that 5 x squared minus 12 x minus 27 equals 0.

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7b
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Solve 5 x squared minus 12 x minus 27 equals 0.
Show all your working and give your answers correct to 2 decimal places.

x equals......................... or x equals......................... 

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7c
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Calculate the perimeter of the triangle.

............................................ m 

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7d
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Calculate the smallest angle of the triangle.

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8
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At a football match, the price of an adult ticket is $x and the price of a child ticket is $left parenthesis x minus 2.50 right parenthesis.
There are 18 500 adults and 2400 children attending the football match.
The total amount paid for the tickets is $320 040.

Find the price of an adult ticket.

$ ................................................ 

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9
Sme Calculator5 marks

Here is a rectangle.

q3-1-hard-2-13-forming-and-solving-equations-edexcel-gcse-maths

The length of the rectangle is 7 cm longer than the width of the rectangle.

4 of these rectangles are used to make this 8-sided shape.

q3-2-hard-2-13-forming-and-solving-equations-edexcel-gcse-maths

The perimeter of the 8-sided shape is 70 cm.
Work out the area of the 8-sided shape.

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10
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A B C D is a rectangle.
E F G H spaceis a trapezium.

q4-hard-2-13-forming-and-solving-equations-edexcel-gcse-maths

All measurements are in centimetres.
The perimeters of these two shapes are the same.

Work out the area of the rectangle.

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11
Sme Calculator4 marks

3 teas and 2 coffees have a total cost of £7.80
5 teas and 4 coffees have a total cost of £14.20

Work out the cost of one tea and the cost of one coffee.

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12
Sme Calculator4 marks

A cinema sells adult tickets and child tickets.

   The total cost of 3 adult tickets and 1 child ticket is £30
   The total cost of 1 adult ticket and  3 child tickets is £22

   Work out the cost of an adult ticket and the cost of a child ticket.

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13a
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The diagram shows a trapezium.

q9a-hard-2-13-forming-and-solving-equations-edexcel-gcse-maths

All the measurements are in centimetres.

The area of the trapezium is 351 cm2.

Show that  2 x squared space plus space x space long dash space 351 equals space 0

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13b
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Work out the value of x.

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1
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Rangan buys 3.6kg of potatoes and 2.8kg of leeks.
The total cost is $13.72.
Leeks cost $2.65 per kilogram.

Find the cost of 1 kg of potatoes.

$ ................................................ 

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2a
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Ahmed sells different types of cake in his shop.
The cost of each cake depends on its type and its size.

Every small cake costs $ x and every large cake costs $ left parenthesis 2 x space plus space 1 right parenthesis.

The total cost of 3 small lemon cakes and 2 large lemon cakes is $12.36.

Find the cost of a small lemon cake.

$ ................................................ 

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2b
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The cost of 18 small chocolate cakes is the same as the cost of 7 large chocolate cakes.

Find the cost of a small chocolate cake.

$ ................................................ [3]

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3
Sme Calculator3 marks

The angles of a quadrilateral are x degree, left parenthesis x plus 5 right parenthesis degree, left parenthesis 2 x minus 25 right parenthesis degree and left parenthesis x plus 10 right parenthesis degree.

Find the value of x.

x = ................................................... 

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4
Sme Calculator3 marks

Oranges cost 21 cents each.
Alex buys x oranges and Bobbie buys left parenthesis x plus 2 right parenthesis oranges.
The total cost of these oranges is $4.20.

Find the value of x.

x = ................................................... 

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5a
Sme Calculator2 marks

Esme buys x magazines at $2.45 each and y cards at $3.15 each.

Write down an expression, in terms of x and y, for the total cost, in dollars, of the magazines and the cards.

$ ................................................... 

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5b
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Esme spends $60.55 in total.
She buys 8 magazines.

How many cards does she buy?

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6
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i)
Factorise  y squared plus 5 y minus 84.

[2]
ii)
cie-igcse-2018-may-jun-3-p4-q5bii
The area of the rectangle is 84 cm2.

Find the perimeter.

.......................................... cm [3]

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7
Sme Calculator4 marks

The diagram shows a trapezium.

xyopd_Z~_q3-meduim-2-13-forming-and-solving-equations-edexcel-gcse-maths

A D = x cm.
B C spaceis the same length as A D.
A B is twice the length of A D.
D C is 4 cm longer than A B.

The perimeter of the trapezium is 38 cm.
Work out the length of A D.

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8
Sme Calculator3 marks

q7-meduim-2-13-forming-and-solving-equations-edexcel-gcse-maths

The diagram shows a parallelogram.
The sizes of the angles, in degrees, are

2 x 3 x space minus 15 2 x 2 x space plus space 24

Work out the value of x.

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9
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The diagram shows a right-angled triangle.

q9-meduim-2-13-forming-and-solving-equations-edexcel-gcse-maths

All the angles are in degrees.
Work out the size of the smallest angle of the triangle.

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10
Sme Calculator5 marks

Redlands School sent x students to a revision day.

   St Samuel's School sent twice as many students as Redlands School.
   Francis Long School sent 7 fewer students than Redlands School.

   Each student paid £ 15 for the revision day.
   The students paid a total of £1155

   Work out how many students were sent by each school to the revision day.
   You must show all your working.

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1
Sme Calculator6 marks

Alan invests $200 at a rate of r percent sign per year compound interest.
After 2 years the value of his investment is $206.46.

i)
Show that     r to the power of 2 space end exponent plus space 200 r space minus space 323 space equals space 0.
[3]
ii)
Solve the equation   r to the power of 2 space end exponent plus space 200 r space minus space 323 space equals space 0  to find the rate of interest.
Show all your working and give your answer correct to 2 decimal places.
 
r space equals space................................................ [3]

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2
Sme Calculator7 marks

Ahmed sells different types of cake in his shop.
The cost of each cake depends on its type and its size.

Every small cake costs $ x and every large cake costs $ left parenthesis 2 x space plus space 1 right parenthesis.

Petra spends $20 on small coffee cakes and $10 on large coffee cakes.
The total number of cakes is 45.

Write an equation in terms of x.
Solve this equation to find the cost of a small coffee cake.
Show all your working.
  

$ ................................................ 

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3
Sme Calculator3 marks

he total perimeter of a semicircle is 19.02 cm.
 
Calculate the radius of the semicircle.
 

................................................ cm

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4
Sme Calculator5 marks

Paddy and Anna each invest $2000 for 5 years.
Paddy earns simple interest at a rate of 1.25% per year.
Anna earns compound interest at a rate of r% per year.
At the end of 5 years, Paddy’s investment is worth the same as Anna’s investment.
 
Calculate the value of r.
 

r equals................................................ 

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5a
Sme Calculator5 marks

All the lengths in this question are in centimetres.

cie-igcse-2020-may-jun-p4-tz3-q5

The diagram shows a shape ABCDEF made from two rectangles.
The total area of the shape is 342 cm2.

Show that x squared plus x minus 72 equals 0.

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5b
Sme Calculator3 marks

Solve  space x squared plus x minus 72 equals 0 space by factorisation.
 

x equals .................... or  x equals.................... 

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5c
Sme Calculator2 marks

Work out the perimeter of the shape ABCDEF.
 

 ............................................ cm 

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5d
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Calculate angle DBC.
 

Angle DBC = ................................................ 

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6
Sme Calculator4 marks

cie-igcse-2019-oct-nov-1-q1e

A, B and C lie on the circle, centre O.
Angle AOC = left parenthesis 3 x plus 22 right parenthesis degree and angle ABC = 5 x degree.

Find the value of x.

x = ................................................... 

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7
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In a shop, the price of a monthly magazine is $m and the price of a weekly magazine is $left parenthesis m minus 0.75 right parenthesis.
One day, the shop receives
•  $168 from selling monthly magazines
•  $207 from selling weekly magazines.
The total number of these magazines sold during this day is 100.

i)
Show that  50 m squared minus 225 m plus 63 equals 0.

[3]
ii)
Find the price of a monthly magazine.
Show all your working.

$ ............................................... [3]

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8
Sme Calculator5 marks

Here are two right-angled triangles.

q2-veryhard-2-13-forming-and-solving-equations-edexcel-gcse-maths

Given that

      tan space e space equals space tan space f

find the value of x.

You must show all your working.

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9
Sme Calculator6 marks

Here is a right-angled triangle.

q6-veryhard-2-13-forming-and-solving-equations-edexcel-gcse-maths

All measurements are in centimetres.
The area of the triangle is 2.5 cm2 .

Find the perimeter of the triangle.
Give your answer correct to 3 significant figures.
You must show all of your working.

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10
Sme Calculator5 marks

There are only r red counters and g green counters in a bag.

A counter is taken at random from the bag.

The probability that the counter is green is 3 over 7

The counter is put back in the bag.

2 more red counters and 3 more green counters are put in the bag.
A counter is taken at random from the bag.

The probability that the counter is green is 6 over 13
Find the number of red counters and the number of green counters that were in the bag originally.

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