NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

Basic
Worksheet 1.1 Real Number System
(Refer to Book 4A Ch1 p. 1.7 – 1.13)
Name: _________________________ Class: ___________

Key Points

1. Consider the following numbers and complete the table below.

3
 3.6, 6.4 ,  ,  0.825, 0, 9 , 11, 5.5 01,  12, 1
4

(a) Integer

(b) Positive integer

(c) Rational number

(d) Irrational number

2. Consider the following fractions.

5 11 4 13 7
,  , ,  ,
12 8 15 9 16
Which can be expressed as terminating decimals?
Which can be expressed as recurring decimals?

(a) Terminating decimal

(b) Recurring decimal

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 NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

1 7 5 7
3. (a) Arrange the numbers 1 , , ,  in ascending order.
3 6 3 5
       
          
       
(b) Among the numbers in (a), convert the largest one into a decimal. _________

4. Convert the following decimals into fractions. (Leave your answers in the simplest form.)
(a) 1.8 (b) 1.0 8

Solution
(a) Let x = 1.8 ,
i.e. x = 1.888 888… ...... (1)
10x = 18.888 888… ...... (2)
(2) – (1): 9x = ________
x = ________
 1.8 = ________

(b) Let x = 1.0 8 ,

i.e. x = ____________ ...... (3)
100x = ____________ ...... (4)

NF 5. Simplify each of the following. State whether the result obtained is rational or irrational.
(a) 2 2  2 = _________ (b) 2 2  2 = ________ It is ( rational
(c) / 2 2  2 ). = ________
irrational
It is (rational / irrational). It is (rational / irrational). It is (rational / irrational).

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 NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

Enhanced
Worksheet 1.1 Real Number System
(Refer to Book 4A Ch1 p. 1.7 – 1.13)
Name: _________________________ Class: ___________

Key Points

Simplify each of the following. State whether the result obtained is rational or irrational. (1 – 6)

NF 1. 4  25 NF 2. 3  48
 

It is (rational / irrational). It is (rational / irrational).

3. 2 +  NF 4. 7  ( 7)2
= 

It is (rational / irrational).
It is (rational / irrational).

2 5 5 8
NF 5.  NF 6.
6 3 2 6
 

It is (rational / irrational). It is (rational / irrational).

3
NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

7. Convert the following decimals into fractions. (Leave your answers in the simplest form.)
(a) 1.0 18 (b) 1.01 8

Solution
(a) Let x = 1.0 18 ,
i.e. x = ____________ ...... (1)
( )x = ____________ ...... (2)

(b)

8. (a) Arrange the decimals 0.123, 0.123 , 0.1 23 , 0.12 3 in descending order.

(b) Among the numbers in (a), convert the largest one into a fraction. (Leave your answer in
the simplest form.)
Solution
(a)
(b)

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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

Basic
Solving Quadratic Equations by the
Worksheet 1.2 Factor Method
(Refer to Book 4A Ch1 p. 1.13 – 1.20)
Name: _________________________ Class: ___________

Key Points
If ( px  r )( qx  s)  0 , where p  0 and q  0 ,

then px  r  0 or qx  s  0
r s
x or x
p q

1. In each of the following, determine whether it is a quadratic equation in one unknown. Put a
‘’ in the appropriate box.
Yes No
(a) x2 + 3x – 1 = 0  
(b) x – 5 = 0  
(c) –x2 = y  
(d) x2 – x  
(e) x(x + 1) = x2  

2. Rewrite the following quadratic equations in general form ax2 + bx + c = 0, where a > 0.
(a) x2 – 4x = 1 (b) 9 + 2x2 = x (c) 7x – 3 = x2
___________________ ___________________ ___________________

3. Determine whether each of the following values is a root of the quadratic equation
x2 + 2x – 3 = 0.
(a) 1 (b) 2
Solution
(a) By substituting x = 1 into (b)
x2 + 2x 3 = 0, we have
L.H.S. = ( )2 + 2( )3
=( )
R.H.S. = ( )
∴ 1 (is / is not) a root of
x2 + 2x – 3 = 0.
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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

4. Determine whether each of the following values is a root of the quadratic equation
4x2 + 8x + 3 = 0.
1 3
(a) (b) 
2 2
Solution
(a) (b)

5. Write down the roots of the following quadratic equations.

(a) x(x – 4) = 0 (b) (x + 2)(x – 3) = 0

Root(s): ___________________ Root(s): ___________________

(c) (2x – 1)(x + 4) = 0 (d) (3x + 1)2 = 0

Root(s): ___________________ Root(s): ___________________

Find the roots of the following quadratic equations. (6 – 13)

6. x2 + 2x = 0 7. 3x2  x = 0
Solution Solution

8. x2 + 4x + 3 = 0 9. x2 + 2x – 15 = 0

Solution Solution

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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

10. x2 + 10x + 24 = 0 11. x2 + x – 12 = 0

Solution Solution

12. 3x2 + 12x  15 = 0 13. x2 + 28x + 24 = 0

Solution Solution

Solve the following quadratic equations. (14 – 17)

14. y2 + 14y + 49 = 0 15. 4y2  12y + 9 = 0

Solution Solution

16. y2 – 16 = 0 17. 25y2  1 = 0

Solution Solution

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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

Enhanced
Solving Quadratic Equations by the
Worksheet 1.2 Factor Method
(Refer to Book 4A Ch1 p. 1.13 – 1.20)
Name: _________________________ Class: ___________

Key Points
If ( px  r )( qx  s)  0 , where p  0 and q  0 ,

then px  r  0 or qx  s  0
r s
x or x
p q

1. Determine whether each of the following values is a root of the quadratic equation 6x – x2 = 5.
(a) 2 (b) 5
Solution
(a) (b)

2. Determine whether each of the following values is a root of the quadratic equation 2x2 = 2  3x.
(a) 2 (b) 5
Solution
(a) (b)

Find the root(s) of the following quadratic equations. (3 – 8)

3. 2x2 – 5x – 3 = 0 4. 2x2 – x – 15 = 0

Solution Solution

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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

5. 3x2 + x – 2 = 0 6. 4x2  2x + 12 = 0

Solution Solution

7 25 2
7. x2  x  2= 0 8. x – 5x + 1 = 0
2 4

Solution Solution

Solve the following quadratic equations. (9 – 12)

9. x 2  6x 10. 8x 2  5x

Solution Solution

11. y2 – 5y = 14 12. s = 2 – 3s2

Solution Solution

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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

Find the roots of the following quadratic equations. (13 – 18)

13. x( x  1)  2 x 2 14. x(x  3) = x  3
Solution Solution

15. z2 + 9(z + 2) = 0 16. 3n2 = 2(n + 4)

Solution Solution

17. x(x – 2) = x + 4 18. (p  1)2 = 1 – 10p

Solution Solution

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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

Basic
Solving Quadratic Equations by the
Worksheet 1.3 Quadratic Formula
(Refer to Book 4A Ch1 p. 1.21 – 1.27)
Name: _________________________ Class: ___________

Key Points
The roots of a quadratic equation ax2  bx  c  0 (where a  0 ) are given by:

 b  b 2  4ac
x (Quadratic formula)
2a

Solve the following quadratic equations by taking square roots. (1 – 4)

1. (x + 1)2 = 25 2. (x – 3)2 = 16
Solution Solution

3. 2(x + 3)2 = 32 4. 4(x – 1)2 = 9

Solution Solution

5. For each of the following quadratic equations ax2 + bx + c = 0, complete the following table.
(Leave your answers in surd form if necessary. If an equation has no real roots, state so.)

a b c The root(s)

 (3)  (3)2  4(1)(1)

(a) x2 – 3x + 1 = 0 1 –3 1 
2(1)

(b) x2 + 2x – 3 = 0

(c) x2 – 12x + 36 = 0

(d) 2x2 – x – 1 = 0

(e) 3x2 + 2x + 1 = 0

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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

Solve the following quadratic equations using the quadratic formula. (Leave your answers in surd
form if necessary. If an equation has no real roots, state so.) (6 – 13)
6. x2 – 13x + 12 = 0 7. x2 + 6x – 4 = 0

Solution Solution

8. y2 – 12y + 9 = 0 9. y2 – 8y – 2 = 0

Solution Solution

10. 2u2 + u + 8 = 0 11. 4v2 – 4v + 1 = 0

Solution Solution

12. 5p2 + 7p – 24 = 0 13. 6q2 – 9q + 7 = 0

Solution Solution

12
NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

Enhanced
Solving Quadratic Equations by the
Worksheet 1.3 Quadratic Formula
(Refer to Book 4A Ch1 p. 1.21 – 1.27)
Name: _________________________ Class: ___________

Key Points
The roots of a quadratic equation ax2  bx  c  0 (where a  0 ) are given by:

 b  b 2  4ac
x (Quadratic formula)
2a

Solve the following quadratic equations by taking square roots. (1 – 2)

1. 4( x  2) 2  3  0 2. (3x  1) 2  5  0
Solution Solution

Solve the following quadratic equations using quadratic formula. (Leave your answers in surd form
if necessary. If an equation has no real roots, state so.) (3 – 6)
3. –x2 – 4x + 10 = 0 4. –2x2 + 5x  7 = 0
Solution Solution

5. 2 – 3y + 2y2 = 0 6. 2y  y 2 1  0
Solution Solution

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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

Solve the following quadratic equations using quadratic formula. (Leave your answers in surd from
if necessary. If an equation has no real roots, state so.) (7 – 10)
5 7
7. v2 – 5v – =0 8. 2 p2  p  0
4 4
Solution Solution

9. 0.1x2 + 0.3x  1 = 0 10. 0.5w2 – 1.2w + 0.8 = 0

Solution Solution

Find the roots of the following quadratic equations using an appropriate method. (Leave your
answers in surd form if necessary. If an equation has no real roots, state so.) (11 – 16)
11. x2 + 12(x + 3) = 0 12. 3y(y + 2) – 4 = 0
Solution Solution

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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

13. 9m2 = 7(6m – 7) 14. (n – 3)(n + 1) + (n + 6) = 0

Solution Solution

1
15. (p + 1)(p – 1) = 2p + 5 16. t(t + 2) = t –
4
Solution Solution

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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

Basic
Solving Quadratic Equations by the
Worksheet 1.4 Graphical Method
(Refer to Book 4A Ch1 p. 1.28 – 1.39)
Name: _________________________ Class: ___________

Key Points
We can solve the quadratic equation ax2  bx  c  0 by the graphical method as follows:

Step 1: Plot the graph of y  ax2  bx  c .

Step 2: Find the roots of ax2  bx  c  0 by reading the x-intercepts of the graph in Step 1.

For each of the following quadratic equations, complete the following table. (1 – 2)

1. y = x2 + x + 1 x –2 –1 0 1 2

2. y = 3x2 – x – 7 x –3 –2 –1 0 1 2 3

In each of the following, plot the graph of the quadratic equation using the given table. (3 – 4)
3. y = x2 – x + 4

x –2 –1 0 1 2
y 10 6 4 4 6

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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

4. y = x2 + 3x + 2

x –4 –3 –2 –1 0 1
y 6 2 0 0 2 6

In each of the following, solve the quadratic equation by using the given graph. (5 – 8)
5. x2 – x – 2 = 0 6. –x2 + 3x = 0

Solution Solution
Root(s): _______________ Root(s): _______________

7. 4x2 + 4x + 1 = 0 8. 3x2 + x + 1 = 0

Solution Solution
Root(s): _______________ Root(s): _______________

17
NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

Enhanced
Solving Quadratic Equations by the
Worksheet 1.4 Graphical Method
(Refer to Book 4A Ch1 p. 1.28 – 1.39)
Name: _________________________ Class: ___________

Key Points
We can solve the quadratic equation ax2  bx  c  0 by the graphical method as follows:

Step 1: Plot the graph of y  ax2  bx  c .

Step 2: Find the roots of ax2  bx  c  0 by reading the x-intercepts of the graph in Step 1.

1. Given the graph of y = x2  2x – 3 from x = 2 to x = 4, solve the equation x2  2x – 3 = 0

graphically.
Solution
The x-intercepts of the graph of
y = x2  2x – 3 are ________ and ________.
Therefore, the roots of
x2  2x – 3 = 0 are ________ and ________.

2. (a) Plot the graph of y = –x2 + 6x – 9 from x = 1 to x = 5.

(b) Hence, solve the equation –x2 + 6x – 9 = 0 graphically.
Solution
(a) x 1 2 3 4 5
y

(b) The x-intercept(s) of the graph of

y = –x2 + 6x – 9 is / are
__________________.
Therefore, the root(s) of
–x2 + 6x – 9 = 0 is / are
__________________.

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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

3. (a) Plot the graph of y = 2x2 – x – 1 from x = –2 to x = 2.

(b) Hence, solve the equation 2x2 – x – 1 = 0 graphically.

Solution
(a) x –2 –1 0 1 2
y

(b)

4. (a) Plot the graph of y = –x2 – 2x – 2 from x = –3 to x = 1.

(b) Hence, solve the equation x2  2x  2 = 0 graphically.

Solution
(a) x
y

(b)

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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

5. In each of the following, the graph of a quadratic equation cuts the x-axis at two points P and
Q. Find the coordinates of P and Q.

(a) y y
(b) y = 2x2 + 3x  5
x
P 0 Q

P Q
x
y = x + 3x  4
2
0

Solution
(a) (b)

6. The figure shows the graph of y = –x2 + x + k. It cuts the x-axis at two
points P and Q, and cuts the y-axis at the point R.
(a) Find the value of k.
(b) (i) Find the coordinates of P and Q.
(ii) Find the length of PQ.

Solution
(a)

(b) (i)

(ii)

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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

Basic
Worksheet 1.5 Problems Leading to Quadratic Equations
(Refer to Book 4A Ch1 p. 1.39 – 1.47)
Name: _________________________ Class: ___________

1. The sum of a positive number x and its square is 2. Find x.

Solution

2. The square of a negative number m is greater than three times of it by 28. Find m.

Solution

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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

3. In the figure, the height and the base of the triangle are (y + 4) cm and y cm
respectively. If the area of the triangle is 30 cm2, find the value of y.

Solution

4. In the figure, the length and the width of the rectangle are
(3w + 1) cm and w2 cm respectively. If the perimeter of the
rectangle is 38 cm, find the dimensions of the rectangle.

Solution

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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

5. Jenny is now twice as old as Janice and Janice is t years old now. The product of their ages is
greater than the sum of their ages by 9. How old is Janice now?

Solution

6. The figure shows a square ring. The length of the outer square is 2 cm

and the length of the inner square is ( + 4) cm. If the area of the

shaded region is 19 cm2, find the value of .

Solution

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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

Enhanced
Worksheet 1.5 Problems Leading to Quadratic Equations
(Refer to Book 4A Ch1 p. 1.39 – 1.47)
Name: _________________________ Class: ___________

1. There are two integers. The larger integer is 3 times the smaller integer. If the two integers are
decreased by 1, their product will be 5. Find the two integers.

Solution

2. The product of two consecutive positive even numbers is greater than three times of their sum
by 90. Find the two numbers.

Solution

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NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

3. The figure shows a rhombus where the lengths of the diagonals

are x cm and (x + 9) cm. If the area of the rhombus is 10 cm2,
find the value of x using the graph below, correct to the nearest
0.2.

Solution

4. A circular flower bed with diameter 14 cm is surrounded by a path

as shown in the figure. If the area of the path is 170 cm2,
(a) express the area of the square ABCD in terms of x,
 22 
(b) find the value of x.  Take   . 
 7 

Solution path

(a)

25
NSS Mathematics in Action (2nd Edition) 4A Section Worksheets 1 Quadratic Equations in One Unknown (I)

(b)

5. A ball is thrown vertically upwards. Its height above the ground after t seconds is given by
(3 + kt – 5t2) m, where k is a constant. It is given that the ball will reach 5 m after 0.4 seconds.
(a) Find the value of k.
(b) When will the ball reach the ground?
(Give your answers correct to 3 significant figures if necessary.)

Solution
(a)

(b)

26