Solve (x 3)(x + 8) = 0
Chapter 1 Quadratic Equations in One Unknown (I)
Question 1
Solve (x ( 3)(x + 8) = 0.
A. x = (3 or 8
B. x = ( 8 or 3
C. x = ( 8 or ( 3
D. x = 3 or 8
Answer: B
Skill: Factor Method
Solution
[pic]
Question 2
The roots of [pic] are and .
Answer: -2, 5 or 5, -2
Skill: Factor Method
Solution
[pic]
∴ The roots of (2 + x)((x + 5) = 0 are (2 and 5.
Question 3
Solve [pic].
A. [pic] or −6
B. [pic] or 23
C. [pic] or −9
D. No real roots
Answer: C
Skill: Factor Method
Solution
[pic]
Question 4
Solve [pic].
A. [pic] or −9
B. [pic]
C. [pic]
D. [pic] or 9
Answer: A
Skill: Factor Method
Solution
[pic]
Try a similar problem:
Solve x(x + 2) = 3.
A. x = –1 or –3
B. x = 1 or –3
C. x = –1 or 3
D. x = 1 or 3
Answer: B
Skill: Factor Method
Solution
[pic]
Question 5
The roots of [pic] are and .
Answer: -11, 2 or 2, -11
Skill: Factor Method
Solution
[pic]
∴ The roots of x2 + 9x – 22 = 0 are (11 and 2.
Question 6
Solve 2x2 ( x (2 = 0.
A. x =[pic] or [pic]
B. x = [pic] or [pic]
C. x = [pic] or [pic]
D. x = [pic] or [pic]
Answer:A
Skill: Quadratic Formula
Solution
2x2 – x – 2 = 0
Using the quadratic formula,
[pic]
Question 7
Solve [pic].
A. [pic] or [pic]
B. [pic] or [pic]
C. [pic] or [pic]
D. [pic] or [pic]
Answer: B
Skill: Quadratic Formula
Solution
[pic]
Using the quadratic formula,
[pic]
Question 8
Which of the following are NOT the roots of [pic]?
I. [pic]
II. [pic]
III. [pic]
A. I and II only
B. II and III only
C. I and III only
D. I, II and III
Answer: B
Skill: Quadratic Formula
Solution
2x2 – 3x – 3 = 0
Using the quadratic formula,
[pic]
∴ [pic] are not the roots of 2x2 – 3x – 3 = 0.
Question 9
Solve [pic].
A. [pic] or [pic]
B. [pic] or [pic]
C. [pic] or [pic]
D. [pic] or [pic]
Answer: A
Skill: Quadratic Formula
Solution
[pic]
Using the quadratic formula,
[pic]
Try a similar problem:
Solve 7x2 = 6x + 3.
A. [pic]
B. [pic]
C. [pic]
D. [pic]
Answer: C
Skill: Quadratic Formula
Solution
[pic]
Using the quadratic formula,
[pic]
Question 10
The roots of [pic] are and . (Give your answers correct to 2 significant figures)
Answer: 0.19, -5.2 or -5.2, 0.19
Skill: Quadratic Formula
Solution
x2 + 5x – 1 = 0
Using the quadratic formula,
[pic]
Question 11
Solve the quadratic equation ax2 + bx + c = 0 using the graph below.
[pic]
A. [pic] or 2.5
B. [pic] or 1
C. [pic] or (1
D. [pic] or (2
Answer: B
Skill: Graphical Method
Solution
The x-intercepts of the graph of y = ax2 + bx + c are (2.5 and 1.
Therefore, the roots of ax2 + bx + c = 0 are (2.5 and 1.
Try a similar problem:
Find the y-intercept of the given quadratic graph shown in the figure.
[pic]
A. (8
B. (9
C. (10
D. (11
Answer: C
Skill: Graphical Method
Solution
∵ The x-intercepts of the given graph are (5 and 2.
∴ The given graph is
[pic]
∴ The y-intercept of the given graph is (10.
Question 12
From the graph below, the root of [pic] is .
[pic]
Answer: 1.5
Skill: Graphical Method
Solution
The x-intercept of the graph of y = 4x2 – 12x + 9 is 1.5.
Therefore, the root of 4x2 – 12x + 9 = 0 is 1.5.
Question 13
The following shows the graph of [pic].
[pic]
How many real roots does the equation [pic] have?
A. 0
B. 1
C. 2
D. Cannot be determined.
Answer: A
Skill: Graphical Method
Solution
∵ The graph of y = px2 + qx + r has no x-intercept.
∴ The equation px2 + qx + r = 0 has no real root.
Question 14
From the graph below, the equation [pic] has real roots. (Input the numbers 0, 1 or 2.)
[pic]
Answer: 2
Skill: Graphical Method
Solution
[pic]
∵ The graph of y = (2x2 + 4x + 3 has two x-intercepts.
∴ The equation 2x2 – 4x – 3 = 0 has two real roots.
Question 15
Form a quadratic equation in x whose roots are greater than the roots of [pic]by 1.
A. 2x2 + 7x + 9 = 0
B. x2 + 3x + 7 = 0
C. x2 + 11x + 18 = 0
D. x2 ( 2x ( 15 = 0
Answer: D
Skill: Forming Quadratic Equation
Solution
[pic]
∴ The roots of the required equation are (4 +1 and 4 + 1, i.e. (3 and 5.
∵ [pic]
∴ [pic]
∴ [pic]
∴ The required equation is x2 – 2x – 15 = 0.
Try a similar problem:
Form a quadratic equation in x whose roots are greater than the roots of
(x – 3)(x + 1) = 0 by 2.
A. x2 ( 6x + 5 = 0
B. x2 + 6x + 5 = 0
C. x2 + 6x ( 5 = 0
D. x2 ( 6x ( 5 = 0
Answer: A
Skill: Forming Quadratic Equation
Solution
[pic]
∴ The roots of the required equation are 3 + 2 and (1 +2, i.e. 5 and 1.
∵ [pic]
∴ [pic]
∴ The required equation is x2 – 6x + 5 = 0.
Question 16
Which of the following quadratic equations has roots [pic] and 2?
A. [pic]
B. [pic]
C. [pic]
D. [pic]
Answer: A
Skill: Forming Quadratic Equation
Solution
The required quadratic equation is
[pic]
Question 17
If the roots of [pic] are [pic] and 3, then [pic] .
Answer: -7
Skill: Forming Quadratic Equation
Solution
The required quadratic equation is
[pic]
∴ [pic]
Question 18
Form a quadratic equation in x whose roots are the reciprocals of the roots of [pic].
A. [pic]
B. [pic]
C. [pic]
D. [pic]
Answer: C
Skill: Forming Quadratic Equation
Solution
[pic]
∴ The roots of the required equation are [pic] and [pic], i.e. [pic] and [pic].
∵ [pic]
[pic]
∴ The required equation is 6x2 + 5x + 1 = 0.
Question 19
The product of two consecutive positive integers is 132. Then the larger integer is
.
Answer: 12
Skill: Application of Quadratic Equation
Solution
Let x be the larger integer, then x – 1 is the smaller integer.
[pic]
∴ The larger integer is 12.
Question 20
The product of two consecutive positive odd numbers is 63. Then the smaller number is .
Answer: 7
Skill: Application of Quadratic Equation
Solution
Let x be the smaller number, then x + 2 be the larger number.
[pic]
∴ The smaller number is 7.
Try a similar problem:
The product of two consecutive positive even numbers is 80. Then the larger number is .
Answer: 10
Skill: Application of Quadratic Equation
Solution
Let x be the larger number, then x – 2 be the smaller number.
[pic]
∴ The larger number is 10.
Question 21
The figure shows a right prism with square base of length x cm. If the height of the prism is 10 cm and the total surface area is 138 cm2, find the value of x.
[pic]
A. 2
B. 3
C. 4
D. 5
Answer: B
Skill: Application of Quadratic Equation
Solution
Total surface area = 138 cm2
[pic]
Question 22
The square of the difference between a number and 6 is equal to the number itself. Then the number is or .
Answer: 9, 4 or 4, 9
Skill: Application of Quadratic Equation
Solution
Let x be the number.
[pic]
∴ The number is 9 or 4.
Question 23
In the figure, the area of △ABC is 30 cm2 and [pic]. Find the value of x.
[pic]
A. 10
B. 8
C. 6
D. 4
Answer: A
Skill: Application of Quadratic Equation
Solution
Area of △ABC = 30 cm2
[pic]
Question 24
In the figure, ABCD is a trapezium with area 24 cm2. Then [pic] .
[pic]
Answer: 4
Skill: Application of Quadratic Equation
Solution
Area of trapezium ABCD = 24 cm2
[pic]
Question 25
Tom is one year older than Kelvin. If the product of their ages is 272, Tom is
years old.
Answer: 17
Skill: Application of Quadratic Equation
Solution
Let x be the age of Tom, then x – 1 is the age of Kelvin.
[pic]
∴ Tom is 17 years old.
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[pic]
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