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For use only in Whitgift School IGCSE Higher Sheets 4

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IGCSE Higher

Sheet H4-1 2-07b-1Completing the Square Sheet H4-2 2-07b-2Completing the Square Sheet H4-3 2-07b-3Quadratic Formula Sheet H4-4 2-07b-4Quadratic Formula Sheet H4-5 2-07c-01 Quadratic Factorisation-Problems Sheet H4-6 2-07c-02 Quadratic Factorisation-Problems Sheet H4-7 2-07c-03 Quadratic Factorisation-Problems Sheet H4-8 2-07c-04 Quadratic Factorisation-Problems Sheet H4-9 2-07c-05 Quadratic Factorisation-Problems Sheet H4-10 2-07c-06 Quadratic Formula-Problems Sheet H4-11 2-07c-07 Quadratic Equations Revision Sheet H4-12 2-07c-08 Quadratic Equations Revision Sheet H4-13 2-07c-09 Quadratic Equations Revision Sheet H4-14 2-07d-01 Quadratic Simultaneous Equations Sheet H4-15 2-07d-02 Quadratic Simultaneous Equations Sheet H4-16 2-08a-1Quadratic Inequalities Sheet H4-17 2-08b-1Graphical Inequalities Sheet H4-18 2-08b-2Graphical Inequalities Sheet H4-19 2-08b-3Graphical Inequalities Sheet H4-20 2-08b-4Graphical Inequalities



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Sheet H4-1 2-07b-1Completing the Square

1. Write the following in the form x + p( )2 + r where p and r are numbers to be determined:

2 2

2 2

2 2

2 2

2 2

(a) 8 7 (b) 12 25(c) 18 75 (d) 6 5(e) 10 7 (f ) 12 3

h(g) 2 1 ( ) 8 3(i) 4 1 ( j) 14 40

x x x xx x x xx x x xx x x xx x x x

+ + + +

+ + + +

+ + + +

+ − + −

+ + + +

2. Write the following in the form x + p( )2 + r where p and r are numbers to be determined:

2 2

2 2

2 2

2 2

2 2

(a) 8 11 (b) 12 13(c) 18 60 (d) 6 6(e) 10 15 (f ) 12 29

h(g) 2 3 ( ) 8 13(i) 4 10 ( j) 14 3

x x x xx x x xx x x xx x x xx x x x

− + − +

− + + +

− − + +

− + + +

− − + −

3. Solve the following equations by completing the square (leaving square roots in your

answers):

2 2

2 2

2 2

(a) 2 1 0 (b) 4 3 0(c) 12 36 0 (d) 20 5 0(e) 8 9 0 (f ) 2 7 0

x x x xx x x xx x x x

+ − = − − =

+ + = + + =

+ − = − − =

4. Solve the following equations by completing the square. First write them in the

form 02 =++ qpxx . Leave square roots in your answers.

2 2

2 2

(a) 2 4 6 0 (b) 3 15 12 0(c) 2 10 1 0 (d) 2 8 12 0

x x x xx x x x

+ − = + − =

+ + = + − =

5. (a) If 2f ( ) 4 5x x x= + + then show that it can be written as ( )2f ( ) 2 1x x= + + .

(b) Use this to explain why )(f x cannot take a value lower than 1. (c) Use this also to explain why )(f x takes the minimum value of 1 when 2x = − .

6. By completing the square, find the minimum value of the following functions:

2

2

2

(a) f ( ) 2 3(b) g( ) 10 3(c) h( ) 4 1

x x xx x xx x x

= + −

= − +

= − +

PTO



Sheet H4-1 2-07b-1Completing the Square (cont.)

7. By completing the square, find the co-ordinates of the turning points of following curves

2

2

2

2

(a) 6 1(b) 2 5(c) 6 9(d) 1

y x xy x xy x xy x x

= + −

= − +

= + +

= + −

8. (a) Find the minimum value of the function 2f ( ) 4 5x x x= + + .

(b) How many solutions are there to the equation 0542 =++ xx ? (c) Explain what happens when you try to solve the equation 0542 =++ xx by completing the square. (d) What is the largest value of p for there to be at least one solution to the equation

042 =++ pxx ?

9. By completing the square on the denominator find the maximum value of the function

2

6f ( )6 11

xx x

=+ +

. What value of x achieves this maximum?



Sheet H4-2 2-07b-2Completing the Square

1. Solve the following quadratic equations by completing the square (leave your answers in the form qp ± ):

2

2

2

2

2

2

(a) 4 1 0(b) 6 7 0(c) 10 20 0(d) 8 11 0(e) 2 7 0(f ) 10 25 0

x xx xz zz zh hc c

+ + =

− + =

− + =

+ + =

+ − =

+ + =

2. Solve the following quadratic equations by completing the square (leave your answers in

the formp q

r±

): 2 2

2 2

2 2

(a) 3 1 0 (b) 5 8 0(c) 7 2 0 (d) 3 0(e) 11 1 0 (f ) 7 3 0


+ − = + − =

+ − = + − =

+ + = + − =

3. Solve 22 5 0x x+ − = by completing the square (leave your answers in the formp q

r±

):

(HINT first divide both sides by 2).

4. Solve 23 7 1 0x x+ + = by completing the square (leave your answers in the formp q

r±

):

(HINT first divide both sides by 3).

5. Find the turning points of the following quadratics by completing the square:

2

2

(a) 2 1(b) 3 7 5

y x xy x x= + −

= + −

6. (a) By completing the square, find the minimum y-value of the curve 742 +−= xxy .

(b) Using part (a), state how many times the curve 742 +−= xxy crosses the line 2=y . (c) Using part (a), state how many times the curve 742 +−= xxy crosses the line 5=y .

PTO



Sheet H4-2 2-07b-2Completing the Square (cont.)

7. Find the equation of the following curves in the form cbxaxy ++= 2 : (a) (b)

-5 -4 -3 -2 -1 1 2 3 4x

-2

2

4

6

8

10

12

14

16

18

y

-5 -4 -3 -2 -1 1 2 3 4 5 6x

-24

-22

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

2

4

6

y

(c) (d)

-3 -2 -1 1 2 3 4 5 6 7x

-18-16-14-12-10-8-6-4-2

2468

101214161820222426283032

y

-7 -6 -5 -4 -3 -2 -1 1 2 3 4x

-55

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

5

10

15

20

25

30

35

y



Sheet H4-3 2-07b-3 Quadratic Formula

The solutions of the equation 02 =++ cbxax are2 4

2b b acx

a− ± −

= .

1. When using the formula

2 42

b b acxa

− ± −= to solve the quadratic equation

02 =++ cbxax , the expression under the square root, i.e. 2 4b ac− , is called the discriminant.

Find the discriminant in the following quadratic equations:

0111)f(0735)e(0372)d(01132)c(0168)b(056)a(

22

22

22

=−−=−−

=++=−+

=+−=++

xxxxxxxx

xxxx

2. Find the solutions to the above quadratic equations, leaving your answers in the form

rqp ±

.

3. Hence solve the quadratic equations in question 1 (to 3sf where necessary). 4. Find the solutions to the following quadratic equations, leaving your answers in the form

rqp ±

.

0811)f(0757)e(

012)d(01135)c(023)b(015)a(

22

22

22

=−−=−+

=−+=−−

=−+=++

xxxxxxxxxxxx

5. Solve the following using the formula (to 3sf), by first writing them in the form r

qp ±:

2 2

2 2

2 2

(a) 3 5 1 0 (b) 7 10 1 0(c) 5 2 4 0 (d) 2 4 0(e) 4 9 3 0 (f ) 13 20 5 0

x x z zc c q qu u t t

+ + = + − =

− − = + − =

+ + = + + =

6. Solve the following equations (to 3sf) using the formula, by first writing them in the form

rqp ±

:

2 2

2 2

(a) 11 1 0 (b) 5 17 53 0(c) 4 19 11 (d) 7 5 13

x x x xx x x x− − = + − =

+ = = +

PTO



Sheet H4-3 2-07b-3 Quadratic Formula (cont.)

7. Solve the following equations (to 3sf) using the formula, , by first writing them in the form

rqp ±

:

( )( )

2 2

2 2

2 2

(a) 11 1 0 (b) 2 9 3 0(c) 3 7 9 (d) 5 6 3(e) 3 4 4 3 6 (f ) 2 1 3 5 7

x x x xx x x xx x x x x x

+ + = + − =

+ = = +

+ + = + + + + =

8. Solve the following equations (to 3sf) using the formula :

1136)d(100117)c(

033125)b(01132)a(22

22

+==+

=−+=−−

xxxxxxxx



Sheet H4-4 2-07b-4Quadratic Formula


2b b acx

a− ± −

= .

1. Solve the following using the formula (to 3sf), making sure that you first of all write the

answers in the form r

qp ±:

2 2

2 2

2 2

(a) 7 5 0 (b) 2 1 0(c) 3 7 0 (d) 5 2 0(e) 4 3 11 0 (f ) 5 7 2 0

x x y ya a h h

d d z z

+ + = + − =

− − = + − =

− − = − − =

2. Solve the following using the formula (to 3sf), making sure that you first of all write the

answers in the form r

qp ±:

2 2

2 2

2

(a) 9 7 8 0 (b) 4 3 11 0(c) 5 7 3 (d) 2 6 3(e) 11 7 10 (f ) (7 1) 5

2 3(g) ( 1)( 3) 7 (h) 34

7 5 2(i) 4 1 ( j) 52 3

2 4(k) 7 (l) 2 103 3

p p k kz z y y

t t u uxq q x

xx x xx x

x xx x

+ − = − − =

− = − =

= + + =+

+ + = =+

+= + + =

+

+ = − =+ −

In the following you need to consider the value of 2 4b ac− in the formula 2 4

2b b acx

a− ± −

= .

3. If the equation 082 =++ cxx (where c is a constant) has two solutions for x then write

down an inequality for c. (Use the formula and consider the expression under the square root).

4. If the equation 092 =++ bxx (where b is a constant) has exactly one solution for x then

find b. 5. If the equation 0182 =++ xax (where a is a constant) has no solutions then write down

an inequality for a.



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Sheet H4-5 2-07c-01 Quadratic Factorisation-Problems

1. Solve the following equations:

2 2

2 2

2 2

2 2

2 2

2 2

(a) 7 10 0 (b) 10 16 0( ) 8 15 0 (d) 2 1 0c(e) 5 14 0 (f ) 3 10 0(g) 10 24 0 (h) 7 18 0(i) 16 0 ( j) 2 0(k) 5 0 (l) 36 0

a a w wx x y yh h r rt t k ky m mu u k

+ + = + + =

− + = − + =

+ − = + − =

− − = + − =

− = + =

− = − =

2. Solve the following equations: ( ) ( )(a) 2 35 (b) 3 28a a c c+ = + =

3. A photograph of area 40 2cm is 3cm longer than it is wide.

If its width is x cm then: (a) Write down an expression for the length in terms of x. (b) Write down an equation which x satisfies and hence show that 04032 =−+ xx . (c) Find the two solutions to this equation. (d) Hence write down the width of the photograph.

4. A rectangle has area 24 2cm its length is 5cm shorter than its width. If its width is w cm then:

(a) Write down an expression for the length in terms of w. (b) Write down an equation which w must satisfy. (c) Find the two solutions to this equation. (d) Hence write down the width of the rectangle.

5. A triangle is such that its width is 4cm longer than its height. The area of the triangle is 30 2cm .

(a) Write down an expression for the width in terms of h where h cm is the height of the triangle.

(b) Write down an equation which h must satisfy. (c) Find the two solutions to this equation. (d) Hence find the height of the triangle.

PTO



Sheet H4-5 2-07c-01 Quadratic Factorisation-Problems (cont.)

6. The triangle and the rectangle have the same area. (a) Write down an equation involving x. (b) Find the two solutions to this equation. (c) Hence write down the one possible value of x in the above diagrams. 7. (a) Solve the equation 0232 =++ xx . (b) Solve the equation 01272 =+− xx .

12 (x+3)

xx




1. A triangle of area 30 2cm is such that is height is 4cm greater than its base. If its base is xcm then: (e) Write down an expression for the height in terms of x. (b) Use the area of the triangle to write down an equation which x satisfies and hence show that 2 4 60 0x x+ − = . (c) Solve this equation to find the base of the triangle.

2. The two rectangles shown below have the same area.

(a) Use the fact that they have the same area to write down an equation involving x. (b) Show that this equation can be written in the form 2 3 10 0x x− − = . (c) Hence find x.

3. A triangle has an area of 52 2cm and its height is 5cm greater than its base. If its base is

xcm then: (a) Write down an expression for the height in terms of x. (b) Hence write down an expression for the area of the triangle in terms of x. (c) Use the fact that the area is 52 2cm to find x. (NB 104 8 13= × )

4. A rectangle measures 4cm by 8cm. It has a strip of xcm placed around it as shown below.

The area of this strip is 64cm2. (a) Write down the dimensions, in terms of x, of the larger rectangle. (b) Hence find the area, in terms of x, of the strip. (c) Use the fact that this area is 64cm2 to find x.

PTO

8cm

4cmx

x

1x −

3x + 1x +

3




5. The sum of the first n positive whole numbers is ( )12

n n +.

(a) Find the sum of the first 20 positive whole numbers. (b) If the sum of the first n numbers is 55 then write down a quadratic equation

involving n. (c) Solve this to find n.

6. A rectangle that measures x by 1x − has the same area as a rectangle that measures 3x −

by 10. (a) Write down a quadratic equation involving x. (b) Solve this to find two possible values of x. 7. Two positive numbers differ by 2. The sum of their squares is 244.

(a) If n is the smaller of the two numbers then write down the larger number in terms of n.

(b) Hence write down an equation involving n. (c) Show that this can be re-written as 2 2 120 0n n+ − = . (d) Solve this to find n.



Sheet H4-7 2-07c-03 Quadratic Factorisation-Problems 1. 28m of fencing is arranged so that it encloses a rectangular area of 40 2m . If w is the width of the rectangle then:

(a) Find the length of the rectangle in terms of w. (b) Write down a quadratic equation involving w. (c) Solve this equation to find w and so find the dimensions of the rectangle.

2. A piece of paper measures 12cm by 8cm. A strip of width xcm is cut off from each side. The area is now 32 2cm .

(a) Find the dimensions of the new piece of paper, in terms of x. (b) Find the area of the new piece of paper, in terms of x. (c) Hence write down a quadratic equation involving x. (d) Show that this equation simplifies to 2 10 16 0x x− + = . (e) Solve this to find the value of x.

3. A right-angled triangle has a width of x. Its height is 7cm more than its width. The hypotenuse is 13cm.

(a) Write down an expression for the height of the triangle in terms of x. (b) Use Pythagoras’s theorem to write down a quadratic equation involving x. (c) Solve this equation to find x.

4. A right-angled triangle has a width of x. Its height is 3cm more than its width. The area is 27 2cm . (a) Write down an expression for the height of the triangle in terms of x. (b) Write down an expression for the area of the triangle in terms of x. (c) Write down a quadratic equation involving x. (d) Solve this equation to find x.

PTO

40 2m

13cm




5. The base of a rectangular box is 3cm longer than twice its width. The area of the base is

44 2cm . Find the width of the box. (Call the width of the box x, create a quadratic equation and solve this).

6. A rectangular carpet measures 2m by 3m. Its width and length are then both increased by x

metres. If its area is now 8.75 2m then: (a) Write down an equation involving x. (b) Show that x satisfies 24 20 11 0x x+ − = (c) Find x.

7. A square is adjoined by four rectangles, each of width 2 units. If the total area is then 33

square units, what is the length of the square? (Call the length of the square x).




1. 50m of fencing is arranged so that it encloses a rectangular area of 154 2m . If w is the width of the rectangle then: (d) Use the fact there is 50m of fencing to find the length of the rectangle in terms of w. (e) Write down a quadratic equation involving w. (f) Solve this equation to find w.

2. Two positive numbers are such that the bigger one is 3 less than twice the smaller one. Their product is 35. If the smaller of the two numbers is x then: (a) Write down an expression for the larger number in terms of x. (b) Write down a quadratic equation involving x. (c) Solve this equation to find x.

3. A triangle whose area is 76 2cm is such that the base is 3cm longer than twice the height. If the height of the triangle if h then: (d) Write down an expression for the base of the triangle in terms of h. (e) Write down a quadratic equation involving h. (f) Solve this equation to find h ( you may use the fact that 19 16 304× = ).

4. A rectangular garden is 7m longer than it is wide and the distance from one corner to the corner diagonally opposite it is 13m. If x is the width of the garden then: (a) Write down an expression for the length of the garden in terms of x (b) Using Pythagoras’ theorem, write down a quadratic equation involving x. (c) Solve this equation to find x.

5. A square piece of paper is trimmed to form a rectangle. One of its side is reduced by 3cm, the other side is reduced by 4cm. The resulting rectangle has an area of 20 2cm . If the side length of the initial square was s then: (a) Write down a quadratic equation involving s. (b) Solve this equation to find s.

h



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1. A rectangular field is 50m longer than it is wide. The area of the field is 2m975 . Find the width of the field. (Call the width of the field x, create a quadratic equation and solve this).

2. The base of a triangle is 4cm longer than its height. If its area is 2cm30 then find the base

of the triangle. Call the height of the triangle x and solve the resulting quadratic equation. 3. Two positive numbers differ by ½. Their product is 68. If the smaller one is x then

establish a quadratic equation and solve it to find the two numbers. 4. A rectangular garden measures 10m by 15m. Its width and length are then both increased

by x metres. If its area is now 2m266 find x (by first creating a suitable quadratic equation).

5. A square is adjoined by two rectangles, each of length 5 units. If the total area is then 39

square units, what is the length of the square? (Call the length of the square x).

6. The sum of the first n numbers 1, 2,…, n is ( )2

1+nn . The sum of the first how many

numbers is 171. (First establish a quadratic equation in n). 7. A rectangular sheet of metal is such that its length is 5cm longer than its width. A square

with side length 3cm is cut out at each corner and the remaining shape is the net of a box. If the volume of this box is 3cm450 find the length of the metal sheet.

PTO




8. A small rectangular field is 5m longer than it is wide. The diagonal of the field is 25m.

Find the dimensions of the field. 9. A man drives x km South and then drives 17km further than that to the West. He ends up

25km away from where he started. What is x 10. A pencil box (in the shape of a cuboid) is 9cm longer than it is wide and 1cm higher than it

is wide. The longest pencil which can be jammed into the case is 13cm long. Find the dimensions of the pencil case.





2b b acx

a− ± −

= .

1. A rectangle is such that its length is 2 metres longer than its width.

(a) If the width of the rectangle is x metres then find expressions for the length and the area of the rectangle in terms of x.

(b) If the area is 5 2m then find x (to 3sf), by solving a quadratic equation.

2. A right angled triangle is such that its hypotenuse is 3 metres longer than twice its shortest side. (a) If the length of the shortest side is x metres and the other side (i.e. not the

hypotenuse) is 4 metres then, by using Pythagoras’ Theorem, write down an equation involving x.

(b) Show that this can be written as 23 12 7 0x x+ − = . (c) Use the formula to find x (to 3sf).

3. A fence is put around a rectangular field which is 50m longer than it is wide. If the width

of the field is x and the area of the field is 56496 2m then: (a) Write down an equation involving x. (b) Calculate x and use this to find how much fencing was required.

4. A rectangle is 3m longer than it is wide. If its area is 2m29 , then find the dimensions of

the rectangle (to 3sf). 5. The shortest side of a right angled triangle is 2cm shorter than its next shortest side. If its

area is 2cm7 then find the shortest side of the triangle (to 3sf). 6. A rectangular field is 25m longer than it is wide. The diagonal of the field is 85m. What is

the width of the field? 7. A farmer has 700m of fence to enclose a rectangular area of 28000m2 . Find the

dimensions of the rectangle (to 3sf). 8. The difference between a positive number and its reciprocal is 4. What is the number (to

3sf)? 9. The sides of a certain rectangle are said to be in the ‘golden ratio’, that is when the largest

possible square is cut off the rectangle, the sides of the remaining rectangle are in the same ratio as the sides of the initial rectangle. Suppose the initial rectangle has a shorter side of 1cm, find the longer side. Hence write down the ‘golden ratio’ (leaving square roots in your answer).

PTO




10. The diagonal of a rectangle is 1cm longer than twice its length. Its width is 7cm. Let x be the length of the rectangle. (a) Write down an expression for the diagonal involving x. (b) Find a quadratic equation involving x by using Pythagoras’s theorem. (c) Solve this to find the length of the rectangle.

11. A rectangular box is 23cm longer than it is wide. Its diagonal is 65cm. If x is the width of

the box then: (a) Find an expression for the length of the box in terms of x. (b) Show that 2 23 1848 0x x+ − = . (c) Solve this equation to find the exact value of x.

12. A rectangular field is 70m longer than it is wide. It is 180m from one corner of the field to the corner diagonally opposite. (a) If x is the width of the field, write down a quadratic equation involving x. (b) Solve this to find x.

13. The rectangular base of a box is such that its length is 17mm longer than its width. The

diagonal is 305mm. Let x be the width of the box. (a) Find an equation involving x. (b) Solve this equation to find x.

14. A rectangle is 34m longer than it is wide. If the diagonal of the rectangle is 50m then:

(a) By letting the width be x, show that 2 34 672 0x x+ − = . (b) Solve this to find the dimensions of the rectangle.

15. (a) Given that the shaded area above is 80 2cm , show that 2 4 83 0x x+ − = . (b) Hence find x to 3sf.

3x +

1x + 3

2



Sheet H4-11 2-07c-07 Quadratic Equations Revision 1. Factorise the following

376)f(495)e(8103)d(992)c(

86)b(158)a(

22

22

22

−−+−

−−++

++++

xxxxxxxx

xxxx

2. Solve the following (without using the formula):

02158)f(072)e(

094)d(0273)c(0145)b(0127)a(

22

22

22

=−−=−

=−=++

=−−=++

xxxxxxx

xxxx

3. Solve the following (to 3sf) by using the formula, making all your working clear:

39)5()f(295)e(

19112)d(0725)c(035)b(0132)a(

2

22

22

=+=+

=+=−+

=−−=−+

xxxxxxxx

xxxx

4. A field is 7m longer than it is wide. The diagonal of the field is 97m. If x is the width of

the field then: (a) Find the length of the field (in terms of x) (b) Find an equation involving x. (c) Solve this equation to find the exact value of x.

5. The length of a rectangle is 5mm longer than its base. The area of the rectangle is 2444 2mm . If x is the base of the rectangle then: (a) Find the length of the rectangle (in terms of x) (b) Find an equation involving x. (c) Solve this equation to find the exact value of x.



Sheet H4-12 2-07c-08 Quadratic Equations Revision

1. Factorise the following:

2 2

2 2

2 2

(a) 6 8 (b) 2 15(c) 12 (d) 9 20(e) 4 (f ) 25

x x x xx x x xx x

+ + − −

+ − − +

− −

2. Factorise the following:

2 2

2 2

(a) 2 7 6 (b) 5 12 4(c) 3 13 10 (d) 16 9

x x x xx x x

+ + − +

− − −

3. Solve the following (without using the formula):

2 2

2 2

(a) 5 4 0 (b) 3 18 0(c) 4 12 7 0 (d) 16 0

x x x xx x x+ + = − − =

+ − = − =

4. Solve the following (to 3sf) by using the formula 2 4

2b b acx

a− ± −

= , making all your

working clear:

2 2

2

(a) 5 2 4 0 (b) 2 5 0(c) 3 1 0 (d) ( 2) 17

x x x xx x x x+ − = + − =

− − = + =

5. A rectangle is 3m longer than it is wide. The diagonal is 11cm. If x is the width of the

rectangle then: (d) Write down the length of the rectangle (in terms of x) (e) Use Pythagoras’ theorem to find an equation involving x and show that it simplifies

to 2 3 56 0x x+ − = (f) Solve this equation to find the value of x (to 3sf).



Sheet H4-13 2-07c-09 Quadratic Equations Revision

1. Factorise the following completely:

2 2 2

2 2 2

2 2 2

(a) 10 21 (b) 16 55 (c) 5 36(d) 7 60 (e) 14 48 (f ) 5 6(g) 6 7 3 (h) 12 3 (i) 45 30 5

x x t t y yr r a a w wb b u q q

+ + + + + −

− − − + − −

+ − − + +

2. Solve the following without using a calculator, showing all your working and giving you

answers as fractions where necessary:

2 2

2 2

2 2

(a) 11 18 0 (b) 4 5 0(c) 14 24 0 (d) 2 1 0(e) 2 3 1 0 (f ) 5 12 4 0

x x x xx x x x

x x x x

+ + = − − =

− + = + + =

+ + = + + =

3. Solve the following equations, by first expressing you answer in the form r

qp ± and

then calculating the solutions to 3sf:

2 2

2 2

2 2

(a) 11 2 0 (b) 5 1 0(c) 10 9 0 (d) 5 17 19 0(e) 2 9 8 0 (f ) 3 5 1 0

x x y yg g t tr r u u

+ + = + + =

+ − = − − =

+ + = − − =

4. A farmer has 220m of fencing to enclose a rectangular area of 2m2500 .

(a) If the width of the field is x then explain why the length is 110 x− . (b) Write down an equation involving x. (c) Solve this to find the dimensions of this field (to the nearest m)?

HARDER QUESTIONS…. 5. A man travels a distance of 300km. On his return journey his average speed was increased

by km/h20 and the time of his journey decreased by 1¼ hr. (a) If v is the average speed of his outward journey then show that

300 300 1.2520v v

− =+

.

(b) Solve this to find v.

6. Solve the following:

( ) ( )

2

2 2

1(a) ( 4) 9 (b) 3

1 1(c) (d) 2 1 1 201

c xx

x x xx

+ = + =

+= + − + =

7. A man has a rectangular sheet of metal. He cuts out the biggest possible circle from this

sheet. With the remaining metal he is then able to cut out another circle which has a radius 5cm smaller than the first. After doing this there is 2cm250 of metal left. Find the radius of the smaller circle.



Sheet H4-14 2-07d-01 Quadratic Simultaneous Equations 1. Solve the following simultaneous equations (by the method of substitution), leaving your

answers as fractions where necessary. All the quadratics can be solved by factorising:

( )

2

2 2

(a) 2 (b) 1 101 2 3

(c) 2 12 (d) 13 2 3 1

(e) 2 2 (f ) 3 2 307 5

xy y xy x y x

x y xy x yy x y x

xy y x x yx y y x

= + =

= + = +

+ = + + = −= − = +

− − = + =+ = = −

2. Solve the following simultaneous equations (by the method of substitution) , leaving your

answers as fractions where necessary:

2 2

2 2 2

(a) 2 (b) 33 3 6

(c) 2 (d) 1692 5 7

y x y xx y y x

y x x yy x y x

= =+ = − =

= + + == + = +

3. Find the point of intersection of the circle and the line shown below:

4. Solve the following simultaneous equations (by the method of substitution):

2 2 2 2

2 2 2 2

(a) 25 (b) 602 10 3 2

(c) 2 144 (d) 2 18 9

x y y xx y y x

x xy y x xy yx y x y

+ = − =+ = = +

+ + = − + =− = + =

2 2 100x y+ =

2y x= +



Sheet H4-15 2-07d-02 Quadratic Simultaneous Equations 1. Solve the following simultaneous equations (by the method of substitution):

2(a) 10 (b) 2 193 2

(c) 16 (d) 2 774 2 1

xy x yy x y x

xy y x xy yy x y x

= + == + = +

+ − = + == + = +

2. Solve the following simultaneous equations (by the method of substitution):

2 2 2 2

2 2 2 2

(a) 25 (b) 607 3 2

(c) 2 144 (d) 2 18 9

x y y xx y y x

x xy y x xy yx y x y

+ = − =+ = = +

+ + = − + =− = + =

3. The ellipse shown below has equation 2 26 2 4 7 0x x y y− + − − = . The straight line has equation 4 1 0y x− − = .

4

4

x

y

Find the points of intersection of the line and the ellipse.

4. Solve the simultaneous equations 2 22 2 7x y x y= − = +



Sheet H4-16 2-08a-1Quadratic Inequalities 1. Solve the following inequalities:

(a) 2 1 5 (b) 3 5 23(c) 9 5 1 24 (d) 24 3 7 66

w xy z

+ ≤ − >< − ≤ < − <

2. Solve the following inequalities:

2 2

2 2

2 2

(a) 1 (b) 16(c) 2 1 17 (d) 3 2 46(e) 28 3 (f ) 15 2 17

x xx x

x x

< >

− ≤ − <

− < − ≤ −

3. Solve the following inequalities (by first factorising the quadratic):

2 2

2 2

2 2

(a) 4 12 0 (b) 6 8 0(c) 5 6 0 (d) 5 6 0(e) 7 12 0 (f ) 3 2 0


− − ≥ + + <

− − ≥ + + <

− + ≥ − + ≤


2

2

2

2

(a) 2 9 7 2 5(b) 3 10 6 3 2(c) 4 17 6 3(d) 3 11 18 2 6

x x xx x xx x xx x x

− + < −

− − < +

+ − < +

− + ≥ +


2

2

2

2

(a) 4 3 4 0(b) 4 6 3 22(c) 11 2 2 10(d) 3 6 8 17

x xx x x

x x xx x

+ − ≤

+ − > − −

− > +

− < −



Sheet H4-17 2-08b-1Graphical Inequalities 1.

(a) Find the equations of the three lines, labelled A, B and C in the above diagram.

These have equations (not in order) 4=y , 43 += xy and xy 29 −= . Write down which line has which equation. Line A =y Line B =y

Line C =y

(b) Leave unshaded the triangular region enclosed between these three lines. (c) Find the three inequalities which define this region (including the three lines).

PTO

Line B Line A

Line C



Sheet H4-17 2-08b-1Graphical Inequalities (cont.)

2.

The above diagram shows the straight line xy 46 −= .

(a) On this diagram draw, and label, the lines 10+= xy and 24 −= xy . (b) Leave unshaded the region satisfied by the four inequalities: xy 46 −≥ , 24 −≥ xy , 0≥x and 10y x≤ + . (c) Write down the coordinates of the four vertices of this unshaded region.

6 4y x= −



Sheet H4-18 2-08b-2Graphical Inequalities 1.

(a) Using dotted lines for all three lines draw, and label, on the diagram shown above,

the three straight lines 183 += xy , 1213 −= xy and 1212 −=+ xy

(b) Leave unshaded the region satisfied by the three inequalities: 1213 −> xy , 183 +≤ xy and 1212 −>+ xy (you may need to turn the dotted lines into solid lines).

PTO



Sheet H4-18 2-08b-2Graphical Inequalities (cont.)

2.

(a) On the above diagram are three lines, labelled A, B and C. Find the equations of these three lines.

(b) Write down the inequalities which define the unshaded triangular region enclosed between these three lines.

Line A

Line C

Line B



Sheet H4-19 2-08b-3Graphical Inequalities 1. (a) On graph paper draw, and label, the following straight lines (use a scale of -8 to 8

with 1cm per unit on both axes): 1 2 8 6y x y x y= + + = =

(b) Leave unshaded the area satisfying the following inequalities: 1 2 8 6y x y x y≥ + + ≥ ≤ 2. (a) On graph paper draw, and label, the following straight lines (use a scale of -2 to 14

on the x-axis and -4 to 12 on the y-axis with 1cm per unit on both axes) (use dotted lines) 10 2 4 3y x y x y= − = − = −

(b) Leave unshaded the area satisfying the following inequalities: 10 2 4 3y x y x y< − < − > − 3. On graph paper (using a scale of -4 to 12 on with 1cm per unit on both axes), leave

unshaded the area satisfying the following inequalities: 3 4 24 2 12 1x y x y x+ > + < ≥ 4. Write down the inequalities which define the unshaded region shown below:



Sheet H4-20 2-08b-4Graphical Inequalities

1. (a) On graph paper draw, and label, the following straight lines (use a scale of -4 to 12 on the x axis and 0 to 10 on the y axis with 1cm per unit on both axes): 3 4 8 4y x x y y x= = + = − =

(b) Leave unshaded the area satisfying the following inequalities: 3 4 8 4y x x y y x≥ ≤ + ≤ − ≤ (c) Calculate the co-ordinates of the vertices of this region. 2. (a) On graph paper draw, and label, the following straight lines, using a scale of 1cm

per unit with both axes ranging from 0 to 10). 10205212 =+=++= yxxyxy (b) Leave unshaded the area satisfying the following inequalities: 102052120 ≤+≥++≤≥ yxxyxyy (c) Write down the co-ordinates of the vertices of this region. 3.

(a) Find the equations of the straight lines labelled A, B, C and D. (b) Find the four inequalities which define the region enclosed between A, B, C and D. (c) What is the name of the shape of this region?

4. Two different types of cakes are produced for a tea party - the fudge cake and the

chocolate cake. Each fudge cake uses 3 eggs and 200g of butter whilst each chocolate cakes uses 4 eggs and 150g of butter. The cook has 48 eggs and 3kg of butter with which to make the cakes. (a) If x is the number of fudge cakes used and y is the number of chocolate cakes used

then write down all the inequalities satisfied by x and y . (b) On graph paper draw a set of axes, both ranging from 0 to 20 (1cm per 2 units) and

leave unshaded the region which contains all the possible values of x and y .

Line A

Line B

Line C

Line D

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