Download Chapter 7 Solving Equations PDF

Number and Algebra  Linear and non-linear relationships
 Solve linear equations involving simple algebraic fractions.
• solve a wide range of linear equations, including those involving one or two
simple algebraic fractions, and check solutions by substitution.
• represent word problems, including those involving fractions, as equations
and solve them to answer the question.
 Solve simple quadratic equations using a range of strategies.
• use a variety of techniques to solve quadratic equations, including grouping,
completing the square, the quadratic formula and choosing two integers with
the required product and sum.
Quadratic - the
highest power
of the variable
is two.
i
u2 + 2P = 2h
Air pressure and the
Bernouilli effect.
s = ut + ½at2
Distance travelled by
accelerating object.
∂u
+ ∇2 u + v( x ) u = 0
∂t
Design of integrated circuits in
mobile phones and computers.
A TASK
A graphics calculator can be used to
solve quadratic equations.
Learn how to use a graphics calculator
to solve quadratic equations such as:
6x2 + 7x − 5 = 0
The solutions are x=0.5, and x = ˉ1.67
A LITTLE BIT OF HISTORY
The Babylonians (about 400 BC) appear to be the
first to solve quadratic equations.
The Hindu mathematician Brahmagupta (598-665
AD) essentially used the quadratic formula to
solve quadratics. Brahmagupta used letters for
variables and considered negative numbers.
al-Khwarizmi (c 820 AD) also essentially used the
quadratic formula to solve quadratics but didn't
consider negative solutions.
ax 2 + bx + c = 0
x=
−b ± b 2 − 4ac
2a
Abraham bar Hiyya Ha-Nasi was the first to
publish a book in 1145 which used the complete
quadratic formula.
87
Solving Linear Equations
To solve an equation is to find the value
of the unknown number (the variable) in
the equation.
The formal way to solve an
equation is to choose to do
the same thing to each side
of the equation until only the
unknown number remains on
one side of the equation.
Exercise 7.1
Solve each of the following equations:
x + 7
=5
x + 7 − 7= 5 − 7
x = ˉ2
c – 4
=7
c – 4 + 4 = 7 + 4
c = 11
1
4
7
x + 8 = 3
t ÷ 4 = 3
z ÷ 7 = 11
d ÷ 3
=5
d ÷ 3 × 3= 5 × 3
d = 15
2 b – 7 = ˉ11
5 a + 11 = 43
8 ˉ2w = 48
3
6
9
+
−
×
÷
is
is
is
is
−
+
÷
×
5y = 25
21m = 105
x – 29 = 41
11
13
15
17
19
21
23
ˉ6(m − 2) = 18
4(x + 2) = 16
ˉ2(f + 5) = 10
4(p − 3) = ˉ20
5(a + 6) = 25
7(x − 3) = ˉ28
ˉ3(x + 5) = 12
ˉ5(2x − 3)= 25
ˉ10x + 15= 25
ˉ10x
= 25 − 15
ˉ10x= 10
x = 10 ÷ ˉ10
x = ˉ1
24
26
28
30
32
34
Check: ˉ5(2×ˉ1 − 3) = ˉ5×ˉ5 = 25 36
3(x − 4) = 12
ˉ2(x − 3) = 14
3(x + 5) = ˉ18
4(2x − 2) = 16
ˉ5(2d + 7) = 65
5(2x − 3) = ˉ15
7(2x − 1) = 21
25
27
29
31
33
35
37
2(2x − 5) = 10
6(2x + 4) = ˉ36
ˉ4(5x − 4) = 24
2(2x − 5) = ˉ10
7(ˉ3x + 2) = 35
3(5x − 4) = ˉ15
5(ˉ5x − 4) = 20
38
39
40
41
42
43
44
45
Check: 3(ˉ2×0.8 − 3) − 4×0.8
3×ˉ4.6 − 3.2 = ˉ17  46
2(x – 3) + 3x = 9
If the unknown occurs
5(x – 3) – 2x = 9
more than once, put
3(x + 2) + 2x = 11
the unknowns together
3(2x – 2) + 4x = 14
(5x−3x=2x).
3x + 2(x + 2) = ˉ11
x + ˉ3(x – 1) = 5
2x + 3(x – 1) + 3x = 13
5x +2(2x + 3) – x + 5 = 35
ˉ9x + 5x + 3(1 – 2x) + 5 = 29
Check: 4(ˉ2 − 3) = 4×ˉ5 = ˉ20 
3(ˉ2x − 3) − 4x = ˉ17
ˉ6x − 9 − 4x = ˉ17
ˉ10x − 9= ˉ17
ˉ10x
= ˉ17 + 9
ˉ10x = ˉ8
x = ˉ8 ÷ ˉ10
x = 0.8
10
12
14
16
18
20
22
The inverse of
The inverse of
The inverse of
The inverse of
3(x + 4) = 21
2(a − 3) = 4
2(x + 3) = ˉ16
3(t − 5) = 12
5(b + 4) = 15
ˉ4(n − 7) = 12
4(n − 7) = 12
4(d − 3) = ˉ20
4d − 12 = ˉ20
4d
= ˉ20 + 12
4d = ˉ8
d = ˉ8 ÷ 4
d = ˉ2
88
5a
= ˉ15
5a ÷ 5 = ˉ15 ÷ 5
a = ˉ3
Solving Linear Equations
5x − 3
= 2x + 9
5x − 3 − 2x = 9
3x − 3 = 9
3x
=9+3
3x = 12
x = 12 ÷ 3
x = 4
Check: (0.36)2 + 8×0.36 − 3 = 0 
x
+ 2 = −5
3
Multiply by the
denominator of
the fraction.
x
a
57 7 + 2 = 3
x
Check: ˉ21/3 + 2 = ˉ7 + 2 = ˉ5 
2x − 3 −
= 2
5
59 6 = 4 −1
x −1
=1
2
x−2 −
= 1
63
2
x+6 −
= 3
65
2
61
2x − 3= ˉ2×5
2x − 3= ˉ10
2x = ˉ10 + 3
2x = ˉ7
x = ˉ7÷2
x = ˉ3.5
Check: (2×ˉ3.5−3)/5 = ˉ10/5 = ˉ2 
67
69
x 2x −
−
= 2
2
3
{×6}
Check: 12/2−2×12/3=6−8 = ˉ2 
2x + 1 x
= −4
3
2
2(2x+1)= 3x − 24
4x + 2= 3x − 24
4x − 3x= −24 − 2
x = ˉ26
If the unknown is on
5x − 8 = 2x + 4
both sides of the =,
2x + 2 = 9 − 5x
put the unknowns
5h + 4 = 4h + 7
together .
6y + 3 = y + 28
5w − 4 = 5 − 4w
4(t − 2) = 2t + 4
3(m + 3) + 2 = 15 − m
4x + 5 − x = 2(3 − x) + 4
−
55 4 + 3 = 2
x + 6= ˉ15
x = ˉ15 − 6
x = ˉ21
3x − 4x= ˉ12
ˉx= ˉ12
x = 12
47
48
49
50
51
52
53
54
71
73
75
77
{×6}
Check: (2×ˉ26+1)/3 = ˉ26/2−4
(ˉ52+1)/3 = ˉ13−4
ˉ17 = ˉ17 
79
81
83
85
2x
+5= 2
3
x −5
=1
2
3x − 2
=3
4
x 3
+ =1
2 5
2x x
− =3
3
2
x x
= +4
2 3
x
2x
+3=
2
3
2x + 1 x
+ =5
3
2
3x − 4 x
= −1
2
5
2x 5 −
+ = 3x
3
2
a
56 5 − 2 = 4
y
−
58 3 − 3 = 1
x
−
60 5 −15 = 3
x+2
=1
3
x −5 −
= 2
64
3
x+6
=2
66
3
62
3a
68 4 − 2 = 3
70
72
74
76
78
80
82
84
86
b−2 −
= 2
3
2b + 4
=2
3
x 1
− =2
3 4
3x x
− =2
2
3
x
x
−1 =
4
3
−
x
3x
+1 =
2
3
3x − 2 x
= +6
2
3
3x − 1 x
= +4
5
2
3x x x
− − =2
2
3 4
Chapter 7 Solving Equations
89
Solving Linear Equations
Each day, in Australia, millions of
real-world problems are solved by
the use of linear equations.
The basic idea is to:
1 Write the formula.
2 Substitute.
3 Solve for the unknown.
Exercise 7.2
The circumference, C, of a circle is given by the formula: C = 2πr, where r is
the radius of the circle. A circular horse yard is to have a circumference
of 56 m. What should be the radius of the horse yard?
Expressions such as C = 2πr
C = 2πr
{write the formula}
are linear because the highest
power of r is 1 (r = r1).
56 = 2πr
{substitute C = 56}
56÷(2π) = r
{inverse of × is ÷}
8.91= r
If you don't use the '(' and ')'
Radius of horse yard = 8.91 m.
on your calculator you will
Check: 2πr = 2π×8.91 = 55.98 m
1
90

get 56÷(2π) wrong.
The circumference, C, of a circle is given by the formula: C = 2πr, where r is
the radius of the circle. A circular horse yard is to have a circumference
of 75 m. What should be the radius of the horse yard?
2
The perimeter, P, of a rectangle is given by P = 2(l + b), where l is the length
and b is the breadth. If the length of a house block is 43 m and the perimeter is
138 m, what is the breadth of the house block?
3
Speed, v, is given by the formula: v = s ÷ t ( v = t ), where s is the distance and t
is the time. If thunder is heard eight seconds after the lightning is seen, how far
away was the lightning (Assume sound travels at 330 m/s)?
4
The sum of the interior angles of a polygon is given by the formula:
S = 90(2n − 4), where n is the number of sides on the polygon.
How many sides in a polygon with an interior angle sum of 540°?
5
The volume, V, of a square based prism is given by the formula: V = w2h,
where w is the width of the base and h is the height of the prism. If the width of
the prism is 4 cm and the volume is 96 cm3, what is the height of the prism?
6
The volume of a cylinder, V, is given by the formula: V = πr2h, where r is
the radius of the base of the cylinder and h is the height of the cylinder. If a
cylinder with a base radius of 6.5 cm has a volume of 1487 cm3, what is the
height of the cylinder?
7
The volume of a cone, a circular based pyramid, is given by the formula:
V=
cone. If a cone has a radius of 1.1 m and a volume of 6.8 m3, what is the height
of the cone?
s
πr 2 h
3
where r is the radius of the base of the cone and h is the height of the
Exercise 7.3
Five times a number increased by four
is divided by three to obtain twentytwo. What is the number?
1
2
3
4
Three times a number increased
by five is divided by two to obtain
thirteen. What is the number?
Four times a number decreased
by six is divided by seven to
obtain twenty-two. What is the
number?
Three consecutive numbers have
a sum of forty-five. What are the
numbers (4,5, and 6 are consecutive
numbers)?
Find three consecutive numbers
whose sum is six times the
first number.
A rectangle has a length which is 5 m
less than twice its width. What is the
width of the rectangle, if the perimeter
is 80 m?
5
A rectangle has a length which is
3 m less than twice its width. What
is the width of the rectangle, if the
perimeter is 42 m?
Let the width be w,
2×w +2×(w − 5)= 80
2w + 2w − 10= 80
4w − 10= 80
4w = 80 + 10
4w = 90
w = 90 ÷ 4
w = 22.5 m (the width)
6
A rectangle has a length twice the
size of the width. What is the
width of the rectangle if the
perimeter is 180 cm?
7
An isosceles triangle has two
angles each four times the size of
the third angle. What is the size of
each angle?
8
The crocodile was described as
being one-third tail, one-quarter
head and with a 200 cm body.
What was the length of the
crocodile?
9
The first side of a triangle is 20 cm
longer that the second side. The
third side is half the length of the
second side. What is the length of
the second side if the perimeter is
300 cm?
Let the number be x,
5x + 4
= 22
3
5x + 4 5x + 4 5x 5x x x = 22×3
= 66
= 66 − 4
= 62
= 62 ÷ 5
= 12.4
Check: (5×12.4 + 4)/3 = 66/3 = 22
Check: 2×22.5+2(22.5−5) = 80


The crocodile was described as being
one-third tail, one-quarter head and with
a 300 cm body. What was the length of
the crocodile?
Let the length of the crocodile be x,
1
1
x + x + 300 = x
3
4
4x + 3x + 3600 = 12x
{×12}
7x + 3600 = 12x
3600 = 12x − 7x
3600 = 5x
10 The difference between two
3600 ÷ 5 = x
numbers is 29. One number is 5
x = 720 (crocodile was 7.2 m) less than three times the other.
Check: 720/3+720/4+300 = 720 
What are the two numbers?
Chapter 7 Solving Equations
91
Solving Quadratics
This is a quadratic
because the highest power
of the unknown, x,is two.
ax2 + bx + c = 0
Thousands of real
life problems can be
written as quadratics.
x2 suggests that there
could be none, one, or
two solutions.
Exercise 7.4
Solve the following quadratics:
x2 + 6x + 9 = 0
(x + 3)2 = 0 {perfect square}
x + 3 = 0 {square root both sides}
x = ˉ3 {inverse of +3}
Check: (ˉ3)2 + 6×ˉ3 + 9 = 0 
x2 − 8x + 16 = 0
(x − 4)2= 0
x − 4 = 0
x = 4
{perfect square}
{square root both sides}
{inverse of −4}
Check: (4)2 − 8×4 + 16 = 0 
1x2 + 2x + 1 = 0
2x2 + 4x + 4 = 0
3x2 + 8x + 16 = 0
4x2 + 10x + 25 = 0
5x2 + 14x + 49 = 0
6x2 − 2x + 1 = 0
7x2 − 6x + 9 = 0
8x2 − 8x + 16 = 0
9x2 − 10x + 25 = 0
10x2 − 18x + 81 = 0
4x2 + 12x + 9
=0
2
(2x) + 2×2x×3 + (3)2 = 0
(2x + 3)2 = 0
2x + 3 = 0
2x
= ˉ3
x = ˉ1.5
114x2 + 12x + 9 = 0
12 9a2 + 6x + 1 = 0
134x2 + 8x + 4 = 0
144x2 − 4x + 1 = 0
15 9y2 + 24y + 16 = 0
16 9x2 − 12x + 4 = 0
17 25x2 − 30x + 9 = 0
x2 + 6x + 8 = 0
2
x + 4x + 2x + 8 = 0 {grouping pairs}
x(x + 4) + 2(x + 4)= 0 {factorising }
(x + 4)(x + 2) = 0 {factorising}
Either x + 4 = 0 or x + 2 = 0
x = ˉ4 or
x = ˉ2
18x2 + 5x + 4 = 0
19x2 + 3x + 2 = 0
20x2 + 5x + 6 = 0
21x2 + 7x + 6 = 0
22x2 + 8x + 12 = 0
23x2 + 8x + 15 = 0
24x2 + 6x + 8 = 0
25x2 + 9x + 14 = 0
{perfect square}
{square root both sides}
{inverse of +3}
{inverse of ×2}
2
Check: 4×(ˉ1.5) + 12×ˉ1.5 + 9 = 0 
Check: (ˉ4)2 + 6×ˉ4 + 8 = 0 
Check: (ˉ2)2 + 6×ˉ2 + 8 = 0 
x2 − 3x − 10 = 0
x2 − 5x + 2x − 10 = 0 {grouping pairs}
x(x − 5) + 2(x − 5)= 0 {factorising }
(x − 5)(x + 2) = 0 {factorising}
Either x − 5 = 0 or x + 2 = 0
x = 5 or
x = ˉ2
Check: (5)2 − 4×5 − 10 = 0 
Check: (ˉ2)2 − 4×ˉ2 − 10 = 0 
92
Solving means find
the values of x that
makes this true.
26x2 + x − 6 = 0
27x2 + 5x − 6 = 0
28x2 − 2x − 8 = 0
29x2 − 5x − 6 = 0
30x2 − 4x − 12 = 0
31x2 − 6x + 9 = 0
32x2 − 10x + 9 = 0
33x2 − 8x + 12 = 0
Use a2 + 2ab + b2 = (a + b)2
to solve quadratics.
Completing the Square
x2 + 6x + 2 = 0
= ˉ2
x2 + 6x
2
x + 6x + 9 = ˉ2 + 9
(x + 3)2 = 7
x + 3 = ±√7
x
=ˉ3±√7
x = ˉ0.35 or x = ˉ5.65
x2 + 6x + 9 = (x + 3)2
Add 9 to both sides to complete the square.
(√7)2 = 7 and (−√7)2 = 7
Which is why there are two solutions:
+√7 or −√7 thus ±√7
Exercise 7.5
Solve the following quadratics by completing the square:
x2 + 8x − 3 = 0
= 3
{move constant term}
x2 + 8x
x2 + 8x + 16 = (x + 4)2
x2 + 8x + 16 = 3 + 16
(x + 4)2 = 19
x + 4
= ±√19
x =ˉ4±√19
x = 0.36 or x = ˉ8.36
{+16 to complete square}
{perfect square}
{square root both sides}
{inverse of +4}
{use calculator}
Check: (0.36)2 + 8×0.36 − 3 = 0 
Check: (ˉ8.36)2 + 8×ˉ8.36 − 3 = 0 
x2 − 10x + 1 = 0
x2 − 10x = ˉ1
{move constant term}
x2 − 10x + 25 = (x − 5)2
x2 − 10x + 25= ˉ1 + 25
(x − 5)2
= 24
x − 5
= ±√24
x
= 5±√24
x = 9.90 or x = 0.10
{+25 to complete square}
{perfect square}
{square root both sides}
{inverse of +4}
{use calculator}
Check: (9.90)2 − 10×9.90 + 1 = 0 
Check: (0.10)2 − 10×0.10 + 1 = 0 
x2 − 3x − 4 = 0
= 4
x2 − 3x
x2 − 3x + (3/2)2 = (x − 3/2)2
x2 − 3x + (1.5)2= 4 + (1.5)2
(x − 1.5)2 = 6.25
x − 1.5
= ±2.5
x
= 1.5±2.5
x = 4 or x = ˉ1
Check: (9.90)2 − 10×9.90 + 1 = 0 
Check: (0.10)2 − 10×0.10 + 1 = 0 
{move constant term}
{3/2 = half middle number}
{complete square, 3/2=1.5}
{perfect square}
{square root both sides}
{inverse of +4}
1x2 + 6x − 1 = 0
2x2 + 2x − 3 = 0
3x2 + 4x − 4 = 0
4x2 + 4x − 6 = 0
5x2 + 6x − 2 = 0
6x2 + 10x − 5 = 0
7x2 + 2x + 1 = 0
8x2 + 6x + 4 = 0
9x2 + 8x + 3 = 0
10x2 + 12x + 2 = 0
11x2 − 2x − 3 = 0
12x2 − 4x − 1 = 0
13x2 − 6x − 5 = 0
14x2 − 8x − 7 = 0
15x2 − 10x − 2 = 0
16x2 − 12x − 4 = 0
17x2 − 14x + 4 = 0
18x2 − 6x + 2 = 0
19x2 − 4x + 1 = 0
20x2 − 8x + 3 = 0
21x2 − 3x − 1 = 0
22x2 + 3x − 3 = 0
23x2 + 5x − 5 = 0
24x2 − 5x − 2 = 0
25x2 + 7x − 4 = 0
26x2 − 7x − 3 = 0
27x2 + 9x + 1 = 0
28x2 − 9x + 2 = 0
29x2 + 11x + 5 = 0
30x2 − 11x + 2 = 0
Chapter 7 Solving Equations
93
The Quadratic Formula
Thousands of problems can be expressed as quadratics. Quadratics can be solved by
guess and check, sketching graphs, factorising, and using the quadratic formula.
Given the quadratic:
The solution is:
ax2 + bx + c = 0
This is the world
famous quadratic
formula.
−b ± b 2 − 4ac
x=
2a
Exercise 7.6
Use the quadratic formula to solve the following quadratics:
x2 + x − 2 = 0
x2 + x − 2 = 0
1 Write the quadratic
ax2 + bx + c = 0
2 Write the general quadratic
a=1 b=1 c=ˉ2
3 Decide values of a, b, c
4 Write the quadratic formula
x=
5 Substitute for a, b, and c
x=
6 Calculate
x=
x=
x=
x=
Check: x=1, 12 + 1 − 2 = 0 
Check: x=ˉ2, (ˉ2)2 + ˉ2 − 2 = 0 = 4 − 4 = 0 
1x2 – x – 2 = 0
2x2 + x – 2 = 0
3x2 – 2x – 3 = 0
4x2 + 2x + 1 = 0
5x2 + 3x + 2 = 0
6x2 + 5x + 6 = 0
7x2 + 4x + 3 = 0
8 x2 – 2x + 1 = 0
9x2 – 3x + 2 = 0
10x2 – 5x + 6 = 0
94
−b ± b 2 − 4ac
2a
1 ± 12 − 4 ×1× − 2
2 ×1
−
−
1± 1+ 8
2
−
1± 9
2
−
1± 3
2
± means + or −
−
−
1− 3
1+ 3
or x =
2
2
x = 1 or x = ˉ2
which is the same as
which is the same as
which is the same as
which is the same as
which is the same as
which is the same as
which is the same as
which is the same as
which is the same as
which is the same as
The numerator needs
brackets if using a
calculator.
(x – 2)(x + 1) = 0
(x – 1)(x + 2) = 0
(x + 1)(x – 3) = 0
(x + 1)(x + 1) = 0
(x + 2)(x + 1) = 0
(x + 2)(x + 3) = 0
(x + 1)(x + 3) = 0
(x – 1)(x – 1) = 0
(x – 1)(x – 2) = 0
(x – 2)(x – 3) = 0
The Quadratic Formula
y = 2x2 + 5x − 7 is called a
quadratic equation because the
highest power of x is 2.
Exercise 7.7
Use the quadratic formula to solve the following quadratics:
5x2 − 2x − 4 = 0
5x2 − 2x − 4 = 0
1 Write the quadratic
ax2 + bx + c = 0
2 Write the general quadratic
a=5 b=ˉ2 c=ˉ4
3 Decide values of a, b, c
4 Write the quadratic formula
x=
−b ± b 2 − 4ac
2a
5 Substitute for a, b, and c
x=
−− 2 ± ( − 2) 2 − 4 × 5 × − 4
2×5
6 Calculate
x=
2 ± 9.17
10
x=
(2 + 9.17)
10
Check: x=1.12, 5×1.122 − 2×1.12 − 4 = 0 
Check: x=ˉ0.72, 5×(ˉ0.72)2 − 2×ˉ0.72 − 4 = 0
See Technology 7.1
or x =
(2 + 9.17)
10
x = 1.12 or x = ˉ0.72

1 2x2 + 3x – 2 = 0
2x2 + 2x – 5 = 0
44x2 – 3x – 3 = 0
3 5x2 – 2x – 1 = 0
6x2 – 3x – 5 = 0
5 2x2 + x – 2 = 0
87x2 – 2x – 5 = 0
7x2 + 5x + 1 = 0
10x2 – 11x – 1 = 0
94x2 + 8x + 3 = 0
124x2 – 5x = 0
11 6x2 + 8x – 4 = 0
Nothing is all wrong.
144x2 – 8 = 0
13x2 + 3x + 2 = 0
Even a busted clock is
16 5x2 – 4x = 0
158x2 + 3x= 0
right twice a day.
1814x2 – 5 = 0
17 16x2 – 4= 0
20 ˉ4x2 – 3x + 1 = 0
19 ˉx2 + 5x – 2 = 0
22 ˉ3x2 + 3x + 1 = 0
21 ˉ2x2 + 3x + 5 = 0
24 ˉ3x2 – 6x – 2 = 0
23 ˉx2 – 4x – 1 = 0
25 A stone is thrown vertically with a velocity of 30 m/s.
The motion of the stone is described by the relationship:
20 m
5t2 - 30t + h = 0, where h is the height in metres and
t is the time in seconds.
How long will it take the stone to reach a height of 20 m?
26 Calculate x in each of the following figures:
Perimeter=60
a)
b)
c)
5
x+2
x
x+2
Area=5
x
x+15
3x
x2
Chapter 7 Solving Equations
95
Mental Computation
Mental computation gives you
practice in thinking.
Exercise 7.8
1 Spell Quadratic.
2 What is the quadratic formula?
3 3x2 + 2x − 2 = 0. What are the values of a, b, and c?
4 Expand (x − 2)2
5 Factorise x2 + 4x + 3
6 What is the value of: Log10100
7 Solve: x + y = 10, xy = 21
8 ˉ2 − ˉ5
6 + 6×30/100
9(x−3)4
= 6 + 180/100
10 Increase $6 by 30%
= 6 + 1.8
x2 + 4x + 3
3+1=4
3×1=3
= (x+3)(x+1)
= $7.80
Exercise 7.9
1 Spell 'Solving linear equations'.
2 What is the quadratic formula?
3 5x2 − x − 3 = 0. What are the values of a, b, and c?
4 Expand (x + 2)2
5 Factorise x2 + 7x + 10
Learning without thought is
labor lost; thought without
6 What is the value of: Log101000
learning is perilous - Confucius.
7 Solve: x + y = 11, xy = 30
8 ˉ2 − 3
9(x−2)4
10 Increase $7 by 30%
War does not determine
who is right, only who is
left - a paraprosdokian.
Exercise 7.10
1 Spell 'Completing the square'.
2 What is the quadratic formula?
3x2 − 2x + 4 = 0. What are the values of a, b, and c?
4 Expand (x − 3)2
5 Factorise x2 + 5x + 4
6 What is the value of: Log1010 000
7 Solve: x + y = 6, xy = 8
You do not need a parachute to skydive.
8 ˉ7 + ˉ7
You only need a parachute to skydive
twice - another paraprosdokian.
9(x−2)5
10 Increase $8 by 30%
Mechanical engineers design and supervise the construction
and operation of machinery.
• Relevant school subjects are English, Mathematics, and Physics.
• Courses usually involve an engineering degree.
96
Build maths muscle and
prepare for mathematics
competitions at the same
time.
Competition Questions
Exercise 7.11
If 48a = b2 and a and b are positive
integers, find the smallest value of a.
48a = b2
16×3a = b2
42×3a = b2
y=
3x − 5
2
thus a = 3
If a and b are positive integers, find the
smallest value of a:
1 28a = b2
2
24a = b2
3
63a = b2
What is the value of x when y = ˉ2?
What is the value of x
when y = ˉ3?
ˉ3×2 = 3x − 5
ˉ6 + 5= 3x
ˉ1 ÷ 3= x
x = ˉ⅓ or ˉ0.33
4
y = 4x + 3
5
y=
2x 1
+
5
3
6
y=
4x + 3
5
(2x − 3)(x + 5) = 0
Either 2x − 3= 0
2x = 3
x = 1.5
Check: (2×ˉ5−3)(ˉ5+5)
Check: (2×1.5−3)(1.5+5)
= ˉ13×0
= 0×6.5
=0
=0
Or
Solve each of the following
quadratic equations:
7 (x + 1)(x −1) = 0
x + 5 = 0
x = ˉ5
8
(x − 2)(2x − 1) = 0
9
(3x + 1)(x − 2) = 0
10 (2x − 2)(x + 4) = 0
11 What is the equation of each of the following quadratics:
x-intercepts
x = ˉ1 thus x + 1 = 0
x = 2 thus x − 2 = 0
5
4
3
2
1
0
-2
-1
-1
0
1
2
3
y = (x + 1)(x − 2)
or y = x2 − x − 2
-2
-3
a)
b)
5
6
5
4
4
3
3
2
2
1
1
-3
0
-4
-3
-2
-1
-1
-2
-3
0
1
2
3
-2
-1
0
-1 0
1
2
3
4
5
-2
-3
-4
-5
Chapter 7 Solving Equations
97
A Couple of Puzzles
Exercise 7.12
1 On Monday a plant is 2 cm high. Each day the plant doubles its
height from the day before. How high will the plant be on Friday?
2 Can you cut this shape into
a) Two conguent shapes?
b) Three conguent shapes?
c) Four congruent shapes?
d) Six congruent shapes?
A Game
Calculator Hi Lo is played on a calculator. You try to guess the secret number in as
few guesses as possible.
1
Someone else gives the limits of the secret
number eg., 1 to 10, 1 to 100, 1 to 1000 and
enters the secret number in the calculator's memory
and clears the display.
2
You enter a guess on the calculator.
3
4
Someone else divides your number by
the number in memory.
If the display is:
greater than 1 then the guess was too high.
less than 1 then the guess was too low.
equal to 1 then the guess was correct.
÷
74
STO
50
RCL
M+
=
13.867321
0.67567567
Try to guess the secret number
in as few guesses as possible.
A Sweet Trick
Impress your audience by showing that you know some tricky patterns.
98
37×3
37×6
37×9
37×12
37×15
=
=
=
=
=
1×8 + 1
=
12×8 + 2
=
123×8 + 3 =
1234×8 + 4 =
12345×8 + 5 =
Think about your presentation:
• Think up some story about
these numbers?
• Audience has calculators,
you don't have a calculator?
92 − 22=
892 − 122=
8892 − 1122=
88892 − 11122=
888892 − 111122=
62 − 52=
562 − 452=
5562 − 4452=
55562 − 44452=
555562 − 444452=
Investigations
Investigation 7.1
A quadratic function
The distance an object falls in a certain time,
when dropped, can be modelled by the
quadratic function: s = 4.9t2,
where s is the distance in metres
an object falls in t seconds.
Investigate
Example: A stone takes 2.75
seconds to reach the bottom of
a well.
s = 4.9t2
Depth of well = 4.9x2.752
= 37 m
Ways of testing the function
A GPS unit and a stone jump off a
cliff at the same time. Which will
hit the ground first?
s = 4.9t2
The GPS because it doesn't have to
ask for directions.
Investigation 7.2
A linear function?
1
Turn on a tap and let the water
run at a constant rate.
2
Ready the bucket by putting a ruler
inside the bucket.
3
Put the bucket under the tap and record the height
of water every 10 seconds (You may wish to record
every 30 s or so dependent upon how much water
is flowing from the tap).
4
Draw a graph.
30
25
20
15
10
5
0
0
1
2
3
4
5
6
You need:
• a bucket
• a ruler
• a stopwatch
Two statisticians were flying from Perth to Sydney.
About an hour into the flight, the pilot announced,
'Unfortunately, we have lost an engine, but don't
worry: There are three engines left. However,
instead of five hours, it will take seven hours to get
to Sydney.'
A little later, the pilot told the passengers that a
second engine had failed. 'But we still have two
engines left. We're still fine, except now it will take
ten hours to get to Sydney.'
Somewhat later, the pilot again came on the
intercom and announced that a third engine had
died. 'But never fear, because this plane can fly
on a single engine. Of course, it will now take 18
hours to get to Sydney.'
At this point, one statistician turned to another and
said, 'Gee, I hope we don't lose that last engine, or
we'll be up here forever!'
Chapter 7 Solving Equations
99
Technology
Technology 7.1
The Quadratic Formula and the Calculator
You can be more efficient and accurate in the use of your calculator.
Example: Solve: 5x2 − 2x − 4 = 0
1 Write the quadratic
2 Write the general quadratic
3 Decide values of a, b, c
5x2 − 2x − 4 = 0
ax2 + bx + c = 0
a=5 b=ˉ2 c=ˉ4
4 Write the quadratic formula
x=
−b ± b 2 − 4ac
2a
5 Substitute for a, b, and c
x=
−− 2 ± ( − 2) 2 − 4 × 5 × − 4
2×5
6 Calculate
x=
2 ± 9.17
10
x=
(2 + 9.17)
10
Use these brackets
on your calculator.
or x =
(2 + 9.17)
10
x = 1.12 or x = ˉ0.72
To calculate
this in one go.
( − 2) 2 − 4 × 5 × − 4
you need an extra
set of brackets.
( ( − 2) 2 − 4 × 5 × − 4 )
√
(
(
±
2
)
x2
−
4
You may need to
practice this on a
calculator.
×
5
×
±
4
)
=
to give the answer 9.17
Technology 7.2
Quadratics and the Graphics Calculator
Use a graphics calculator to check your answers to Exercises 7.4, 7.5, 7.6, and 7.7.
Example:
Graph: y = x2 + x − 6 and solve: x2 + x − 6 = 0
Press
Y=
and enter the function eg., x2 + x − 6
Press
Graph
to see a graph of the function.
To solve: x2 + x − 6 = 0 is to find the x-intercepts.
a)Use
CALC
to find the intercepts (some calculators use zero and value).
b)Use TRACE and move the cursor to the x-intercepts.
c)Use
100
TABLE
to find the x-intercept (where y = 0).
Can you get the answers:
x = ˉ3 and x = 2
Chapter Review 1
Exercise 7.13
1
2
3
4
5
6
7
x 2x −
−
= 2
2
3
3x − 4x= ˉ12
ˉx= ˉ12
x = 12
{×6}
Check: (ˉ13+3)/5 = ˉ10/5 = ˉ2 
3(x − 4) = 15
ˉ2(x − 3) = 8
7(2x − 1) = 14
2(x – 3) + 3x = 11
5(x – 3) – 2x = 12
6x − 8 = 2x + 4
2x + 2 = 16 − 5x
8
x
+ 5 = −1
3
9
2x −1
=2
3
10
3x x
− =7
2
3
11 The sum of the interior angles of a polygon is given by the formula:
S = 90(2n − 4), where n is the number of sides on the polygon.
How many sides in a polygon with an interior angle sum of 540°?
12 The perimeter of a block of land is 110 m. The length is 21 m longer than the
breadth. Find the length and the breadth.
x2 − 8x + 16 = 0
(x − 4)2= 0
x − 4 = 0
x = 4
{perfect square}
{square root both sides}
{inverse of −4}
Check: (4)2 − 8×4 + 16 = 0 
x2 − 3x − 10 = 0
x − 5x + 2x − 10 = 0 {grouping pairs}
x(x − 5) + 2(x − 5)= 0 {factorising }
(x − 5)(x + 2) = 0 {factorising}
Either x − 5 = 0 or x + 2 = 0
x = 5 or
x = ˉ2
2
Check: (5) − 4×5 − 10 = 0 
Check: (ˉ2)2 − 4×ˉ2 − 10 = 0 
2
25x2 + 5x + 2 = 0
26 3x + 7x − 3 = 0
Given the quadratic:
2
27x2 – 11x + 3 = 0
28 ˉ4x2 – 3x + 1 = 0
29
The solution is:
13x2 + 4x + 4 = 0
14x2 + 10x + 25 = 0
15x2 − 2x + 1 = 0
16x2 − 8x + 16 = 0
17x2 + 5x + 4 = 0
18x2 + 8x + 12 = 0
19x2 + 5x − 6 = 0
20x2 − 2x − 8 = 0
21x2 − 5x − 6 = 0
22x2 − 4x − 12 = 0
23x2 − 10x + 9 = 0
24x2 − 8x + 12 = 0
ax2 + bx + c = 0
x=
−b ± b 2 − 4ac
2a
A stone is thrown vertically with a velocity of 40 m/s. The motion of the stone
is described by the relationship: 5t2 - 40t + h = 0, where h is the height in
metres and t is the time in seconds. How long will it take the stone to reach a
height of 10 m?
30 Calculate x in the
following figure:
7
x+1
x
Chapter 7 Solving Equations
101
Chapter Review 2
Exercise 7.14
x 2x −
−
= 2
2
3
3x − 4x= ˉ12
ˉx= ˉ12
x = 12
{×6}
Check: (ˉ13+3)/5 = ˉ10/5 = ˉ2 
1
2
3
4
5
6
7
2(x − 3) = 10
ˉ5(x − 4) = 10
4(3x − 1) = 20
3(x – 1) + 2x = 12
7(x – 2) – 3x = 4
7x − 7 = 2x + 3
2x + 2 = 13 − 2x
8
x
− 2 = −5
5
9
3x − 2
=5
2
10
2x x
− =5
3
4
11 The volume, V, of a square based prism is given by the formula: V = w2h,
where w is the width of the base and h is the height of the prism. If the width of
the prism is 6 cm and the volume is 162 cm3, what is the height of the prism?
12 An isosceles triangle has two angles each twice the size of the third angle.
What is the size of each angle?
x2 − 8x + 16 = 0
(x − 4)2= 0
x − 4 = 0
x = 4
{perfect square}
{square root both sides}
{inverse of −4}
Check: (4)2 − 8×4 + 16 = 0 
x2 − 3x − 10 = 0
x − 5x + 2x − 10 = 0 {grouping pairs}
x(x − 5) + 2(x − 5)= 0 {factorising }
(x − 5)(x + 2) = 0 {factorising}
Either x − 5 = 0 or x + 2 = 0
x = 5 or
x = ˉ2
2
Check: (5) − 4×5 − 10 = 0 
Check: (ˉ2)2 − 4×ˉ2 − 10 = 0 
2
25x2 + 3x + 1 = 0
26 3x + 5x − 2 = 0
Given the quadratic:
2
27x2 – 7x + 5 = 0
28 ˉ2x2 – x + 1 = 0
29
ax2 + bx + c = 0
x=
−b ± b 2 − 4ac
2a
A stone is thrown vertically with a velocity of 50 m/s. The motion of the stone
is described by the relationship: 5t2 - 50t + h = 0, where h is the height in
metres and t is the time in seconds. How long will it take the stone to reach a
height of 40 m?
30 Calculate x in the
following figure:
102
The solution is:
13x2 + 6x + 9 = 0
14x2 + 8x + 16 = 0
15x2 − 4x + 4 = 0
16x2 − 10x + 25 = 0
17x2 + 4x + 3 = 0
18x2 + 6x + 8 = 0
19x2 + x − 6 = 0
20x2 − 3x − 10 = 0
21x2 − 5x − 6 = 0
22x2 − x − 12 = 0
23x2 − 9x + 8 = 0
24x2 − 13x + 12 = 0
x+1
Area=4
x