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π

CHAPTER 2

Quadratic functions

Quadratic Functions
Quadratic functions are in the form of 𝒇(𝒙) = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄, where 𝒂 ≠ 𝟎 and
represented by:

Vertex (minimum point)
axis of symmetry

Vertex (minimum point)
𝒂<𝟎

𝒂>𝟎

Roots of a quadratic function:
→ The roots of the function are the x-coordinates of the points of intersection of the
curve with the x-axis
...
53

−𝑏±√𝑏2 −4𝑎𝑐
2𝑎

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CHAPTER 2

Types of roots of a quadratic function:
1) Two different (distinct) real roots:

2) Two equal real roots (one real roots):

3) No real roots:

P
...

Discriminant = 𝛥 = 𝑏 2 − 4𝑎𝑐, we have three cases:
1) 𝛥 = 𝑏 2 − 4𝑎𝑐 > 0

"two distinct "("different")" real roots"

2) 𝛥 = 𝑏 2 − 4𝑎𝑐 = 0

"two equal real roots "("one root")

3) 𝛥 = 𝑏 2 − 4𝑎𝑐 < 0

"no real roots"

Examples:
1) State the nature of the roots for each of the following quadratics:
a) 𝑓(𝑥) = 𝑥 2 − 10𝑥 + 25
∵𝑎=1

𝑏 = −10

𝑐 = 25

∴ 𝛥 = 𝑏 2 − 4𝑎𝑐 = 100 − 100 = 0

It has one real root (two equal roots)

-------------------------------------------------------------------------------------------------------------------------

b) 𝑓(𝑥) = −𝑥 2 + 5𝑥 + 6
∵ 𝑎 = −1

𝑏=5

𝑐=6

∴ 𝛥 = 𝑏 2 − 4𝑎𝑐 = 25 + 24 = 49 > 0

It has two different roots (two distinct roots)

-------------------------------------------------------------------------------------------------------------------------

c) 𝑓(𝑥) = −2𝑥 2 − 5𝑥 − 6
∵ 𝑎 = −2

𝑏 = −5

𝑐 = −6

∴ 𝛥 = 𝑏 2 − 4𝑎𝑐 = 25 − 48 = −23 < 0

It has no real roots

P
...

∵𝑎=3

𝑏=2

𝑐=𝑘

∵ 𝛥 = 𝑏 2 − 4𝑎𝑐 = 0

two equal roots
4

1

∴ 4 − 4 × 3 × 𝑘 = 0 → 4 − 12𝑘 = 0 → 12𝑘 = 4 → 𝑘 = 12 = 3
-------------------------------------------------------------------------------------------------------------------------

3) The equation 𝑘𝑥 2 − 2𝑥 − 7 = 0 has two distinct real roots, find the possible value
of k
...
56

−4
28

→ 𝑘>

−1
7

π

CHAPTER 2

Quadratic functions

 Test yourself:
1) Solve the equation 2𝑥 2 − 2𝑥 − 1 = 0, giving the roots in exact form

------------------------------------------------------------------------------------------------------------------------

2) Solve the equation 3𝑥 2 − 4𝑥 − 9 = 0, giving your answers to 2 d
...


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3) State the nature of the roots for each of the following quadratics:
a) 2𝑥 2 − 3𝑥 − 4 = 0

-------------------------------------------------------------------------------------------------------------------------

b) 2𝑥 2 − 3𝑥 − 5 = 0

P
...


P
...


Example:
Find the coordinates of the vertex of the function𝑓(𝑥) = −2𝑥 2 − 3𝑥 + 4; hence,
state the maximum or minimum value of f(x) and the value of x at which it occurs
...
59

3
4

41
8

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CHAPTER 2

Quadratic functions

2) Put the function 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 in the form of the completing the square
𝑓(𝑥) = 𝐴(𝑥 + 𝐵)2 + 𝐶
...
The vertex is (-B, C)
...


Solution
−2𝑥 2 − 3𝑥 + 4 ≡ 𝐴(𝑥 + 𝐵)2 + 𝐶 ≡ 𝐴(𝑥 2 + 2𝐵𝑥 + 𝐵 2 ) + 𝐶 ≡ 𝐴𝑥 2 + 2𝐴𝐵𝑥 + 𝐴𝐵 2 + 𝐶

𝐴 = −2
−3

3

2𝐴𝐵 = −3 ⇒ 2(−2)𝐵 = −3 ⇒ −4𝐵 = −3 ⇒ 𝐵 = −4 = 4
3

𝐴𝐵 2 + 𝐶 = 4 ⇒ −2(4)2 + 𝐶 = 4 ⇒
3 2

∴ 𝑓(𝑥) = −2 (𝑥 + 4) +

41

,

8

−18
16

9

+𝐶 =4⇒ 𝐶 =4+8 =
3 41

the vertex is (− 4 ,

8

41
8

)

-------------------------------------------------------------------------------------------------------------------------

2) The quadratic 𝑥 2 − 10𝑥 + 7 is denoted by f(x)
...
Hence, find the least possible value of f(x) and the corresponding value
of x
...
60

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CHAPTER 2

Quadratic functions

3) The equation of a curve is 𝑦 = 8𝑥 − 𝑥 2
...

b) Find the coordinates of the vertex and state whether it is maximum or minimum
...
61

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CHAPTER 2

Quadratic functions

5) If 𝑓(𝑥) = 2𝑥 2 − 8𝑥 + 10
i) Express 𝑓(𝑥) is the form 𝑎(𝑥 + 𝑏)2 + 𝑐, where a, b and c are constant
ii) hence, state the coordinates of the stationary point of 𝑓(𝑥) and state its type
...


Solution
𝑥 2 + 4𝑥 + 3 ≡ (𝑥 + 2)2 − 4 + 3
𝑥 2 + 4𝑥 + 3 ≡ (𝑥 + 2)2 − 1,

𝑎 = 2 , 𝑏 = −1

-----------------------------------------------------------------------------------------------------------------------

7) Put 2 + 2𝑥 − 𝑥 2 in the form ℎ − (𝑥 + 𝑝)2 stating the values of h and p
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62

ℎ = 3 𝑎𝑛𝑑 𝑝 = −1

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CHAPTER 2

Quadratic functions

8) Express 2𝑥 2 − 16𝑥 + 37 in the form𝐴(𝑥 + 𝐵)2 +𝐶, stating the values of A, B
and C
...


Solution
−2𝑥 2 + 4𝑥 = −2(𝑥 2 − 2𝑥) = −2[(𝑥 − 1)2 − 1]
= −2(𝑥 − 1)2 + 2 = 2 − 2(𝑥 − 1)2

𝐴 = 2, 𝐵 = 2 𝑎𝑛𝑑 𝐶 = 1
P
...
64

π

CHAPTER 2

Quadratic functions

 Test yourself:
1) Given that 𝑥 2 − 4𝑥 + 7 ≡ (𝑥 − 𝑎)2 + 𝑏
...


-------------------------------------------------------------------------------------------------------------------------

2) Express each of the following in the form of (𝑥 + 𝑎)2 + 𝑏; stating the value of a
and b
...

a) 𝑥 2 + 2𝑥 + 2

------------------------------------------------------------------------------------------------------------------------

b) 𝑥 2 − 8𝑥 − 3

P
...
66

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CHAPTER 2

Quadratic functions

g) 7 − 8𝑥 − 4𝑥 2

------------------------------------------------------------------------------------------------------------------------

h) −𝑥 2 − 10𝑥 + 7

------------------------------------------------------------------------------------------------------------------------

3) Find the least or the greatest value of each of the following quadratic and the value
of x for which this occurs
...
67

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CHAPTER 2

Quadratic functions

b) 𝑦 = (𝑥 + 2)2 − 7

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c) 𝑦 = 1 + (2𝑥 − 3)2

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d) 𝑦 = (5𝑥 + 3)2 + 2

------------------------------------------------------------------------------------------------------------------------

e) 𝑦 = 3 − 2(𝑥 − 4)2

P
...

→ Find the y-intercept (value of y when x = 0)
...

→ Sketch the function
...
69

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CHAPTER 2

Quadratic functions

b) 𝑦 = 7 − 10𝑥 − 𝑥 2

Solution
Roots ⇒ 7 − 10𝑥 − 𝑥 2 = 0 ⇒ 𝑥 2 − 10𝑥 + 7
by using the formula ⇒ 𝑥 = −10
...
657
𝑦-intercept ⇒ 𝑥 = 0 ⇒ 𝑦 = 7 ⇒ (0 , 7)
Vertex ⇒ 𝑥 =

−𝑏
2𝑎

−−10

= 2×−1 = −5

,

𝑦 = 7 − 10(−5) − (−5)2 = 32 ⇒ (−5 , 32)

P
...
71

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CHAPTER 2

Quadratic functions

3) 𝑓(𝑥) = 4𝑥 − 2𝑥 2 + 3

------------------------------------------------------------------------------------------------------------------------

4) 𝑓(𝑥) = 𝑥 2 + 6𝑥

P
...

→ Solve the equation (find the roots)
...

→ State the range of values of x satisfying the inequality
...
73

,

𝑥=3

3

x

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CHAPTER 2

Quadratic functions

c) 𝑥 2 + 12 < 13𝑥

Solution
1

12

𝑥 2 − 13𝑥 + 12 < 0
𝑥 2 − 13𝑥 + 12 = 0 ⇒ (𝑥 − 1)(𝑥 − 12) = 0 ⇒ 𝑥 = 1

,

x

𝑥 = 12

∴ 1 < 𝑥 < 12
------------------------------------------------------------------------------------------------------------------------

d) 𝑥 2 > 𝑥

Solution
0

𝑥2 − 𝑥 > 0
𝑥 2 − 𝑥 = 0 ⇒ 𝑥(𝑥 − 1) = 0 ⇒ 𝑥 = 0
∴𝑥<0

and

,

x

1

𝑥=1

𝑥>1

------------------------------------------------------------------------------------------------------------------------

2) Find the range of values of k for which the equation 𝑘𝑥 2 + 𝑘𝑥 + 2 = 0 has no real
roots
...
74

𝑘=8

0

8

k

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CHAPTER 2

Quadratic functions

3) Find the range of values of k for which the equation 𝑘𝑥 2 + 3𝑥 + 𝑘 = 0 has two
distinct real roots
...


Solution
𝑎=1

,

𝑏=𝑘

,

2

𝑐 = 2𝑘 − 3

6

𝛥 = 𝑏 2 − 4𝑎𝑐 ≥ 0 ⇒ 𝑘 2 − 4(1)(2𝑘 − 3) ≥ 0 ⇒ 𝑘 2 − 8𝑘 + 12 ≥ 0
𝑘 2 − 8𝑘 + 12 = 0 ⇒ (𝑘 − 2)(𝑘 − 6) = 0 ⇒ 𝑘 = 2
∴𝑘≤2

,

𝑘≥6

P
...
76

π

CHAPTER 2

Quadratic functions

d) 4 − 9𝑥 2 ≤ 0

------------------------------------------------------------------------------------------------------------------------

2) Find the set of values of k for which the equation 𝑥 2 − 2𝑘𝑥 + 4 = 0 has two real
roots
...
77

π

CHAPTER 2

Quadratic functions

Simultaneous Equations
How to solve a pair of simultaneous equations one of them quadratic and
the other linear:
Examples:
1) Solve simultaneously 𝑥 2 + 2𝑦 2 = 9,

𝑥 + 4𝑦 = 9

Solution
From the line ⇒ 𝑥 = 9 − 4𝑦
Into the curve ⇒ (9 − 4𝑦)2 + 2𝑦 2 = 9 ⇒ 81 − 72𝑦 + 16𝑦 2 + 2𝑦 2 = 9
18𝑦 2 − 72𝑦 + 81 − 9 = 0 ⇒ 18𝑦 2 − 72𝑦 + 72 = 0 → (÷ 18)
𝑦 2 − 4𝑦 + 4 = 0 ⇒ (𝑦 − 2)(𝑦 − 2) = 0 ⇒ 𝑦 = 2
𝑥 = 9 − 4(2) = 1
∴ The line cuts the curve at point (1 , 2)
∴ The line is tangent to the curve and the point of tangency is (1 , 2)
------------------------------------------------------------------------------------------------------------------------

2) Find the point(s) of intersection of the line 𝑥 + 𝑦 = 1 and the curve
𝑥 2 − 𝑥𝑦 + 𝑦 2 = 7

Solution
From the line ⇒ 𝑦 = 1 − 𝑥
Into the curve ⇒ 𝑥 2 − 𝑥(1 − 𝑥) + (1 − 𝑥)2 = 7 ⇒ 𝑥 2 − 𝑥 + 𝑥 2 + 1 − 2𝑥 + 𝑥 2 = 7
3𝑥 2 − 3𝑥 + 1 − 7 = 0 ⇒ 3𝑥 2 − 3𝑥 − 6 = 0 → (÷ 3)
𝑥 2 − 𝑥 − 2 = 0 ⇒ (𝑥 − 2)(𝑥 + 1) = 0 ⇒ 𝑥 = 2
𝑦 = 1 − 2 = −1

,

𝑦 = 1 − −1 = 2

∴ The line cuts the curve at points (2, -1) and (-1, 2)
∴ The points of intersection are (2, -1) and (-1, 2)

P
...

1) 𝛥 > 0

The line cuts the curve at two different (distinct) points
...


3) 𝛥 < 0

The line neither cuts nor touches the curve

Examples:
1) State the relation between the line 𝑥 + 𝑦 = 1and the curve 𝑦 = 𝑥 2 + 2𝑥 − 3

Solution
From the line ⇒ 𝑦 = 1 − 𝑥
Into the curve ⇒ 1 − 𝑥 = 𝑥 2 + 2𝑥 − 3 ⇒ 𝑥 2 + 3𝑥 − 4 = 0
𝑎=1

,

𝑏=3

,

𝑐 = −4

𝛥 = 𝑏 2 − 4𝑎𝑐 = 32 − 4(1)(−4) = 9 + 16 = 25 > 0
∴ The line cuts the curve at two different points
...
79

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CHAPTER 2

Quadratic functions
1

2) Prove that the line 𝑦 = 𝑥 − 1 is a tangent to the curve 𝑦 = 4 𝑥 2
...

------------------------------------------------------------------------------------------------------------------------

3) Find the range of the value of k for which the line 𝑦 − 𝑥 = 1 cuts the curve
𝑦 = 𝑘𝑥 2 at two distinct points
...


Solution
From the line ⇒ 𝑦 = 𝑥 + 𝑘
Into the curve ⇒ 𝑥 2 + 𝑥(𝑥 + 𝑘) + 2 = 0 ⇒ 𝑥 2 + 𝑥 2 + 𝑘𝑥 + 2 = 0 ⇒ 2𝑥 2 + 𝑘𝑥 + 2
=0
𝑎=2

,

𝑏=𝑘

,

𝑐=2

𝛥 = 0 ⇒ 𝑏 2 − 4𝑎𝑐 = 0 ⇒ 𝑘 2 − 4(2)(2) = 0 ⇒ 𝑘 2 − 16 = 0 ⇒ 𝑘 2 = 16
𝑘=4

,

𝑘 = −4
P
...


Solution
From the line ⇒ 𝑦 = 2𝑥 + 𝑐
Into the curve ⇒ (2𝑥 + 𝑐)2 = 4𝑥 ⇒ 4𝑥 2 + 4𝑐𝑥 + 𝑐 2 = 4𝑥 ⇒ 4𝑥 2 + 4𝑐𝑥 − 4𝑥 + 𝑐 2 = 0
4𝑥 2 + (4𝑐 − 4)𝑥 + 𝑐 2 = 0
𝑎=4

,

𝑏 = 4𝑐 − 4

,

𝑐 = 𝑐2

𝛥 = 0 ⇒ 𝑏 2 − 4𝑎𝑐 = 0 ⇒ (4𝑐 − 4)2 − 4(4)𝑐 2 = 0 ⇒ 16𝑐 2 − 32𝑐 + 16 − 16𝑐 2 = 0
16

1

−32𝑐 + 16 = 0 ⇒ 32𝑐 = 16 ⇒ 𝑐 = 32 ⇒ 𝑐 = 2

P
...

Find the coordinates of the points of intersection
...


P
...


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5) Show that the line 𝑦 = 3𝑥 − 3 and the curve 𝑦 = (3𝑥 + 1)(𝑥 + 2) do not meet
...
83

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CHAPTER 2

Quadratic functions

Equations which could be reduced to quadratics:
Examples:
1) solve 𝑥 4 − 4𝑥 2 + 3 = 0

Solution
Let ℎ = 𝑥 2 so

ℎ2 = 𝑥 4

ℎ2 − 4ℎ + 3 = 0 ⇒ (ℎ − 3)(ℎ − 1) = 0 ⇒ ℎ = 3
∴ 𝑥2 = 3

,

𝑥2 = 1

∴ 𝑥 = ±√3

,

,

ℎ=1

𝑥 = ±1

------------------------------------------------------------------------------------------------------------------------

2) Solve 𝑥 − 5√𝑥 = 6

Solution
Let ℎ = √𝑥 so

ℎ2 = 𝑥

ℎ2 − 5ℎ − 6 = 0 ⇒ (ℎ − 3)(ℎ − 2) = 0 ⇒ ℎ = 3
∴ √𝑥 = 3

,

√𝑥 = 2

∴𝑥=9

,

,

ℎ=2

𝑥=4

------------------------------------------------------------------------------------------------------------------------

3) Solve 𝑥 6 − 3𝑥 3 + 2 = 0

Solution
Let ℎ = 𝑥

3

so

2

ℎ =𝑥

6

ℎ2 − 3ℎ + 2 = 0 ⇒ (ℎ − 2)(ℎ − 1) = 0 ⇒ ℎ = 2
∴ 𝑥3 = 2

,

𝑥3 = 1

3

∴ 𝑥 = √2

,

,

ℎ=1

𝑥=1

-----------------------------------------------------------------------------------------------------------------------18

1

4) Find the real roots of the equation 𝑥 4 + 𝑥 2 = 4

Solution
Multiply the equation by 𝑥 4
Let ℎ = 𝑥 2 so ℎ2 = 𝑥 4

18 + 𝑥 2 = 4𝑥 4 ⇒ 4𝑥 4 − 𝑥 2 − 18 = 0

9
, ℎ = −2
4
𝑥 2 = −2 (rejected no square root for − 𝑣𝑒 numbers)

4ℎ2 − ℎ − 18 = 0 ⇒ (4ℎ − 9)(ℎ + 2) = 0 ⇒ ℎ =
∴ 𝑥2 =

9
4

,

P
...
85

π

CHAPTER 2

Quadratic functions

Exercises
1) i) Express 2𝑥 2 + 8𝑥 − 10 in the form 𝑎(x + b) 2 + 𝑐
...


P
...
Express 𝑓(𝑥) in the
form 𝑎(x − b) 2 − 𝑐
...


-------------------------------------------------------------------------------------------------------------------------

4) Determine the set of values of k for which the line 2𝑦 = 𝑥 + 𝑘 does not intersect
the curve 𝑦 = 𝑥 2 − 4𝑥+7
...
87

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CHAPTER 2

Quadratic functions

5) Find the set of values of m for which the line 𝑦 = 𝑚𝑥 + 4 intersects the curve
𝑦 = 3𝑥 2 − 4𝑥 + 7 at two distinct points
...

i) Express 8𝑥 − 𝑥 2 in the form 𝑎 −(x + b) 2 , stating the numerical values of a and b

-------------------------------------------------------------------------------------------------------------------------

ii) Hence, or otherwise, find the coordinates of the stationary point of the curve
...
88

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CHAPTER 2

Quadratic functions

iii) Find the set of values of x for which 𝑦 ≥ −20
...


-------------------------------------------------------------------------------------------------------------------------

8) Find the set of values of k for which the line 𝑦 = 𝑘𝑥 − 4 intersects the curve
𝑦 = 𝑥 2 − 2𝑥 at two distinct points
...
89

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CHAPTER 2

Quadratic functions

9) The function f is defined by 𝑓 ∶ 𝑥 → 𝑥 2 − 3𝑥 for 𝑥 ∈ 𝑅
...


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