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�3 7.1 Solving simultaneous equations
239A
1 x + y = 3 x − y = 1
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4 5x + 2y = 2 2x + 3y = −8
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7 3x + y = 11 y = 2x − 4
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2 3x − 2y = −3 x + 2y = 7
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5 3x − 2y = 4 2x + 3y = −6
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8 2x + y = 0
3 = x + 2y
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3 4x − 3y = 1
x − 2y = 4
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6 x + y = 3 y = x + 1
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9 3x + 2y = 10 4x − y = 6
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Guided practice worksheet
Solve these simultaneous equations.
B
Questions are targeted at the grades indicated
�3 7.1 Solving simultaneous equations
239B
1 0 x + y = 4 y = 2x − 5
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1 2 3x + 7y = 1 2x + 5y + 1 = 0
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1 4 8x + 4y = 7
6x − 8y = 41
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1 1 5x = 2y −14
x − 5y = −12
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1 3 4x + 3y = 4
2x = 5y + 15
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1 5 5x + 10y = 28 15x = 20y − 121
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Guided practice worksheet
B
A
�3 7.2 Setting up equations in two unknowns
241A
1 The sum of two numbers is 17 and twice the larger number exceeds three times the smaller number by 4. Find the value of each of the numbers.
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3 The diagram shows an isosceles triangle.
All the angles are measured in degrees.
Find the size of each angle.
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2 Six pencils and three rubbers cost £6.15
Five pencils and two rubbers cost £4.90
Work out the cost of 12 pencils and 10 rubbers.
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4 A nut and a bolt have a mass of 98 g.
The mass of 4 bolts and 2 nuts is 336 g.
Find the mass of a nut and the mass of a bolt.
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Guided practice worksheet
3p + q
3p – 2q
2p + 5q
A
Questions are targeted at the grades indicated
�3 7.2 Setting up equations in two unknowns
241B
Guided practice worksheet
5 The average of two mixed numbers is 12.
One of the numbers is 3 more than the other.
Find the value of each number.
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7 In my of� ce, I have x two-drawer � ling cabinets and y three-drawer � ling cabinets.
Each drawer of a two-drawer filing cabinet holds 60 files.
Each drawer of a three-drawer filing cabinet holds 50 files.
I have a total of 840 files and all my filing cabinets are full.
a Show that 4x + 5y = 28
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6 Strawberry jam costs 20p per pot more than a pot of marmalade.
Two pots of strawberry jam and three pots of marmalade cost £7.65
Work out the cost of a pot of marmalade.
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I have 2 more three-drawer filing cabinets than two-drawer filing cabinets in my office.
b Work out the number of each type of filing cabinet.
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A
�3 7.3 Using graphs to solve simultaneous equations
243A
1
a The line A has equation x + y = 1
Write down the equation of each of the lines B and C.
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b Write down the pair of simultaneous equations that have a solution.
i x = −1, y = 2
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ii x = 2, y = −1
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iii x = −2, y = 1
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2 For each of these pairs of simultaneous equations, draw two linear graphs on the same grid and use them to solve the simultaneous equations.
a x + y = 3
3x − y = 1
1
0–1–2–3–4 1 2 3 4 5 x
y
–1
–2
–3
–4
2
3
4
C
BA
–1
1
2
3
4
5
y
–2
–3
0 x–3 –2 –1 1 2 3 4 5
Guided practice worksheet
B
Questions are targeted at the grades indicated
�3 7.3 Using graphs to solve simultaneous equations
243B
b x − y = 1
x + 2y = 7
c y = 2x + 2
3x + 2y = 4
d x + 3y = 0
x − 3y = 6
e 3y = 2x + 8
x + y = 1
–1
1
2
3
4
5
y
–2
–3
0 x–3 –2 –1 1 2 3 4 5 6 7 8
–1
1
2
3
4
5
y
–2
–3
0 x–3 –2 –1 1 2 3 4 5
Guided practice worksheet
–1
1
2
3
4
5
y
–2
–3
0 x–3 –2 –1 1 2 3 4 5 6 7 8
–1
1
2
3
4
5
y
–2
–3
–4
–5
0 x–3–4–5 –2 –1 1 2 3 4 5
B
1 Solve
a x(x + 8) = 0 b (x − 4)(x + 2) = 0
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c (2x − 5)(3x − 12) = 0
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2 Solve
a x2 + 4x = 0 b x2 − 5x = 0
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c 3x2 − 6x = 0
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3 Solve
a x2 − 8x + 15 = 0 b x2 + 4x + 3 = 0
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c x2 + x − 20 = 0 d x2 − x − 42 = 0
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e x2 − 8x + 16 = 0 f x2 + 2x + 1 = 0
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4 Solve
a x2 − 49 = 0 b x2 − 400 = 0
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�3 7.4 Solving quadratic equations by factorisation
245
Hint If the product of 2 expressions is zero, then one of them must equal zero.
Guided practice worksheet
B
Questions are targeted at the grades indicated
1 Expand and simplify.
a (x + 2)2 = (x + 2)(x + 2) = ....................................................
b (x + 5)2 = .................................................... = ....................................................
c (x + 8)2 = .................................................... = ....................................................
d (x – 1)2 = .................................................... = ....................................................
2 Use your answers to Question 1 to complete the following.
a x2 + 4x = (x + 2)2 – .................................................... b x2 + 10x = (x + 5)2 – ....................................................
c x2 + 16x = (x + 8)2 – .................................................... d x2 – 2x = (x – 1)2 – ....................................................
3 a Expand and simplify (x + 6)2.
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b x2 + 12x + 39 = (n + p)2 + q
Find the values of p and q.
p = .................................................... q = ....................................................
4 a Expand and simplify (x – 9)2.
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b x2 – 18x + 70 = (x – c)2 – d
Find the values of c and d.
c = .................................................... d = ....................................................
5 a Write x2 + 20x + 140 in the form (x + e)2 + f.
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b State the values of e and f.
e = .................................................... f = ....................................................
6 Write x2 + 22x + 1 in the form (x + g)2 + h.
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�3 7.5 Completing the square
247
Guided practice worksheet
C
A
�3 7.6 Solving quadratic equations by completing the square
249A
1 Here is a diagram for x2 + 10x
By completing the diagram for the part of the square that equates to x2 + 10x,
express x2 + 10x in the form (x + p)2 + q.
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2 Using the method of completing the square, solve these quadratic equations, giving your solutions in surd form.
a x2 − 4x − 2 = 0 b x2 + 8x + 1 = 0
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c x2 + 12x − 4 = 0 d x2 − 6x − 8 = 0
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e 2x2 − 2x − 1 = 0 f 2x2 + 8x + 5 = 0
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x
x ?
x² ?
? ?25
Guided practice worksheet
A
Questions are targeted at the grades indicated
�3 7.6 Solving quadratic equations by completing the square
249B
3 Using the method of completing the square, solve these quadratic equations, giving your solutions correct to 2 decimal places.
a x2 + 6x + 3 = 0 b x2 − 10x + 5 = 0
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c x2 + 2x − 4 = 0 d 2x2 + 8x − 7 = 0
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e 6x2 − 6x − 1 = 0 f 10x2 − 2x − 9 = 0
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Guided practice worksheet
A
�3 7.7 Solving quadratic equations using the formula
251A
For each of these quadratic equations,
a work out the value of b2 − 4ac, and
b use the quadratic formula x = − ± −b b ac
a
2 42 , to solve them.
Give your solutions correct to 3 significant figures.
Guided practice worksheet
Hint The whole of the numerator is divided by 2a.
1 x2 + 5x + 3 = 0
a ................................................
b
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4 x2 − x − 8 = 0
a ................................................
b
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7 2x2 + 7x + 1 = 0
a ................................................
b
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1 0 3x2 − 3x − 7 = 0
a ................................................
b
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2 x2 + 10x + 7 = 0
a ................................................
b
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5 x2 − 5x − 4 = 0
a ................................................
b
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8 5x2 + 9x + 1 = 0
a ................................................
b
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1 1 10x2 + x − 4 = 0
a ................................................
b
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3 x2 + x − 5 = 0
a ................................................
b
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6 x2 − 9x + 4 = 0
a ................................................
b
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9 4x2 − 10x + 3 = 0
a ................................................
b
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1 2 2x2 − 11x + 6 = 0
a ................................................
b
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A
Questions are targeted at the grades indicated
�3 7.7 Solving quadratic equations using the formula
251B
Which of the following quadratic equations have
a two solutions, b one solution, c no solution?
1 3 x2 + 5x + 1 = 0 .....................................................................................................................................................................................................
1 4 x2 + x + 7 = 0 .....................................................................................................................................................................................................
1 5 x2 + 4x + 4 = 0 .....................................................................................................................................................................................................
1 6 x2 − x + 8 = 0 .....................................................................................................................................................................................................
1 7 x2 − 7x − 4 = 0 .....................................................................................................................................................................................................
1 8 2x2 − 3x + 4 = 0 .....................................................................................................................................................................................................
Guided practice worksheet
A
�3 7.8 Solving algebraic fraction equations leading to quadratic equations
253A
1 Solve these fraction equations.
a 1 12
1x x
+ = b 1 14
1x x
− =
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c 4 11
1x x
++
= d x
x x− −23
2 5=
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e 2
23
4 12xx
x− += f
32
22 3
17x x+
−−
=
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Guided practice worksheet
A*
Questions are targeted at the grades indicated
�3 7.8 Solving algebraic fraction equations leading to quadratic equations
253B
Guided practice worksheet
2 Show that the equation xx
−−
47
23 1
= can be written as 3x2 − 13x − 10 = 0 and hence solve xx
−−
47
23 1
=
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3 Solve these fraction equations. Give your solutions correct to 3 signi� cant � gures.
a 11
41
1x x+
+−
= b 3 2 4xx
− =
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c 2 11
3x x
−+
= d xx
x−
−2
3 2=
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e 23
32
1x x x−
−−
= f 22 1
12 1
5x x−
−+
=
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A*
�3 7.9 Setting up and solving quadratic equations
255A
1 The formula S n n= 12 1( )+ gives the sum S of the numbers 1, 2, 3, etc up to n. Find the value of n if S = 136.
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2 Find the number such that twice its square is 11 more than 21 times the original number.
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3 Find the ages of a lady and her daughter if the lady was 23 years old when her daughter was born, and the product of their ages is now 174.
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4 The area of an oil painting is 120 cm2.
The width is 2 cm less than the length.
Find the length of the picture.
Give your answer correct to 2 decimal places.
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Guided practice worksheet Questions are targeted at the grades indicated
A*
�3 7.9 Setting up and solving quadratic equations
255B
5 A garage is 9 metres long.
The area of the 3 rectangular walls is 40 m2. The height is 2 m less than the width.
Find the height of the garage.
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6 A rectangular lawn measures 20 m by 15 m.
The lawn is surrounded by a path of uniform width. If the area of the path is 156 m2, find its width.
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7 The perimeter of a right-angled triangle is 56 cm.
The length of the hypotenuse is 25 cm. Calculate the lengths of the other two sides.
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8 Michael is training for a long-distance cycle race.
One day he cycles for x hours in the morning, travelling a distance of 84 km. In the afternoon, he cycles for 1 hour more travelling the same distance, but his average speed is 2 km/h slower than in the morning.
Calculate the number of hours that Michael cycles in the afternoon.
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Guided practice worksheet
A*
�3 7.10 Constructing graphs of simple loci
257A
1 Use squared paper for your answers.
a Draw the locus of points 4 units from the
i x-axis ii y-axis.
b Write down the equations for each locus.
i .........................................................................................
ii .........................................................................................
2 a Use squared paper to draw each of the following lines.
i x = 6 ii y = 6 iii x = –4 iv y = –5
–1
y
–2
–3
–4
0 x–3–4–5 –2 –1 1 2 3 4 5
–6
–7
–8
–9
–10
–5
–6–7–8–9–10 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10
b Use dotted lines to draw the locus of points 4 units from each of the above lines.
c Write down the equations of each locus.
i .........................................................................................
ii .........................................................................................
iii .........................................................................................
iv .........................................................................................
Guided practice worksheet
–1
1
2
3
4
5
y
–2
–3
–4
–5
0 x–3–4–5 –2 –1 1 2 3 4 5
C
B
Questions are targeted at the grades indicated
�3 7.10 Constructing graphs of simple loci
257B
3 The graph shows the line with equation x + y = 5.
The dotted lines are the locus of points 3 units from the line.
a Use Pythagoras’ theorem to calculate the distance a.
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b Find the equations of the locus.
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4 a Use the graph opposite to draw a circle of radius 6 with the origin (0, 0) as centre.
b On the same axes, draw the line with equation x + y = 0.
c Calculate the equations of the two tangents to the circle that are parallel to x + y = 0.
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d Draw the tangents.
Guided practice worksheet
0
5
3
x + y = 5
a
5
y
x
–1
y
–2
–3
–4
0 x–3–4–5 –2 –1 1 2 3 4 5
–6
–7
–8
–5
–6–7–8 6 7 8
1
2
3
4
5
6
7
8
B
A
Hint The angle between a tangent and a radius of a circle is 90°.
�3 7.10 Constructing graphs of simple loci
257C
5 a Use graph paper to draw the locus of points 7 units from the origin.
–1
y
–2
–3
–4
0 x–3–4–5 –2 –1 1 2 3 4 5
–6
–7
–8
–9
–10
–5
–6–7–8–9–10 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10
b Complete the equation of the locus: x2 + y2 = ..................................................................
c Draw the tangent to the circle at i (7, 0) ii (0, –7)
d Write down the equation of each tangent.
i ..................................................................
ii ..................................................................
6 Find the equations of the locus of points 6 units from the line x + y = 10.
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Guided practice worksheet
Hint Draw a circle with centre at the origin.
A
A*
�3 7.11 Solving simultaneous equations when one is linear and the other is quadratic
259A
For each of these pairs of simultaneous equations,
a draw a quadratic graph and a linear graph on the same grid and use them to solve the simultaneous equations
b solve them using an algebraic method. Give your answers correct to 2 decimal places where appropriate. You must show all of your working.
1 x2 + y = 6
y = −x
a x = ........................................
y = ........................................
b ..............................................
2 x2 + 3y = 7
y = x + 1
a x = ........................................
y = ........................................
b ..............................................
–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9–10 –9 –8 –7 –6 –5 –4 –3 –2 –1–1
1
2
3
4
5
6
7
8
9
10
y
–2
–3
–4
–5
–6
–7
–8
–9
–10
0 1 2 3 4 5 6 7 8 9 10 x
–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9–10 –9 –8 –7 –6 –5 –4 –3 –2 –1–1
1
2
3
4
5
6
7
8
9
10
y
–2
–3
–4
–5
–6
–7
–8
–9
–10
0 1 2 3 4 5 6 7 8 9 10 x
Guided practice worksheet Questions are targeted at the grades indicated
A
�3 7.11 Solving simultaneous equations when one is linear and the other is quadratic
259B
3 y = 1 − 2x2
y = 3 − 4x
a x = ........................................
y =........................................
b ..............................................
4 3y = 5x2 + 8
y = 3 − 4x
a x = ........................................
y = ........................................
b ..............................................
Guided practice worksheet
–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9–10 –9 –8 –7 –6 –5 –4 –3 –2 –1–1
1
2
3
4
5
6
7
8
9
10
y
–2
–3
–4
–5
–6
–7
–8
–9
–10
0 1 2 3 4 5 6 7 8 9 10 x
–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9–10 –9 –8 –7 –6 –5 –4 –3 –2 –1–1
1
2
3
4
5
6
7
8
9
10
y
–2
–3
–4
–5
–6
–7
–8
–9
–10
0 1 2 3 4 5 6 7 8 9 10 x
A
�3 7.12 Solving simultaneous equations when one is linear and one is a circle
261A
1 On graph paper, draw the graph of the circle with equation x2 + y2 = 4.On the same axes, draw the straight line with equation y = x.
Hence find estimates of the solutions of the simultaneous equations x2 + y2 = 4 and y = x.
2 Draw suitable graphs to � nd estimates of the solutions of the simultaneous equations
x2 + y2 = 9 and y = x – 2.
3 Draw suitable graphs to � nd estimates of the solutions of the simultaneous equations
x2 + y2 = 100 and 2x + 3y = 12.
Guided practice worksheet
–1
1
2
3
4
5
y
–2
–3
–4
–5
0 x–3–4–5 –2 –1 1 2 3 4 5
–1
1
2
3
4
5
y
–2
–3
–4
–5
0 x–3–4–5 –2 –1 1 2 3 4 5
–1
y
–2
–3
–4
0 x–3–4–5 –2 –1 1 2 3 4 5
–6
–7
–8
–9
–10
–5
–6–7–8–9–10 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10
A*
Questions are targeted at the grades indicated
�3 7.12 Solving simultaneous equations when one is linear and one is a circle
261B
4 Solve algebraically these simultaneous equations.
Give your answers correct to 3 significant figures where appropriate.
a x2 + y2 = 25 b x2 + y2 = 10 c x2 + y2 = 40
y + x = 7 y = 1 − x y = 2 + 3x
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5 Solve algebraically these simultaneous equations.
Give your answers correct to 3 significant figures.
a x2 + y2 = 32 b x2 + y2 = 50 c x2 + y2 = 75
y = 2x + 1 y = 2 − 3x y = 4x − 6
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Guided practice worksheet
A*
