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Mathematics LibreTexts

Common Mistakes

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The purpose of this page is to alert the reader to the common mistakes which might be made in simple calculations. The reader should care fully analyze each statement, making sure that they can justify the correct statement.

Answer true or false for each of the statements given below. If the statement is false, write the correct version.

Exercise 1

1. (a+b)=a+b

Answer

False. Leave a+b as is. It cannot be simplified any further.

Exercise 2

2. a1/2=1a2

Answer

False. a1/2=1a

Exercise 3

3. (a+b)2=a2+b2

Answer

False. (a+b)2=a2+2ab+b2

Exercise 4

4. (2)2=4

Answer

False. (2)2=4

Exercise 5

5. 34=81

Answer

False. 34=(34)=81

Exercise 6

6. 23=6

Answer

False. 23=8

Exercise 7

7. a+bc=ac+bc

Answer

True.

Exercise 8

8. ca+b=ca+cb

Answer

False. ca+b cannot be simplified any further.

Exercise 9

9. 5hh(h+2)=4hh+2

Answer

False. 5hh(h+2)=5h+2

Exercise 10

10.03 is not defined.

Answer

False. 03=0

Exercise 11

11. 50=0

Answer

False. 50 is undefined.

Exercise 12

12. xy2=x2y2

Answer

False. xy2 cannot be simplified any further.

Exercise 13

13. 2x=0x=2

Answer

False. x=0

Exercise 14

14. 3x=2x=2+3=5

Answer

False. x=23

Exercise 15

15. x2+1=0x=±1

Answer

False. x2+1=0 does not have any real solutions. It does have complex solutions x=±i.

Exercise 16

16. (x2)2=0x=±2

Answer

False. x=2

Exercise 17

17. The solutions of the quadratic equation c2+5c+6=0 are given by c=±(5c6).

Answer

False. Factor the quadratic (c5)(c+1)=0. Then c=5,1.

Exercise 18

18. (c1)(c3)=1c1=1, or c3=1

Answer

False. This is true if (c1)(c3)=0 (not 1).

Exercise 19

19. 2(43)=86

Answer

False. 243=83

Exercise 20

20. 123=32

Answer

False. 123=16

Exercise 21

21. The only solution of the quadratic equation x22x=0 is x=2.

Answer

False. x=0 is also a solution.

Exercise 22

22. x1x+5=0x1=0, or x+5=0

Answer

False. Only x1=0 (if x+5=0, this would cause division by zero).

Exercise 23

23. 3x2=32x

Answer

True.

Exercise 24

24. By factoring an expression such as x24, we mean: Solve x24=0 which yields x2=4x=±2

Answer

False. By factoring an expression such as x24, we mean: x24=(x2)(x+2).

Exercise 25

25. The conjugate of (x+1)+2 is (x1)2.

Answer

False. The conjugate is x+12.

Exercise 26

26. ((x+1))2=(x+1)2

Answer

False. (x+1)2=x+1

Exercise 27

27. x1 is the same as 1x

Answer

False. x1 is the same as (1x).

Exercise 28

28. (x1)(x3)(3x)=x1

Answer

False. (x1)(x3)3x=(x1),x3

Exercise 29

29. x1=(x1)

Answer

False. x1=(x+1)

Exercise 30

30. 4x+4x+8=4+49=89

Answer

False. 4x+4x+8=4(x+1)x+8, which cannot be simplified any further.

Exercise 31

31. 3hh=2h

Answer

False. 3hh=3,h0

Exercise 32

32. (x25x+1)=x25x+1

Answer

False. (x25x+1)=x2+5x1

Exercise 33

33. 13x+37x=47x

Answer

False. 13x+37x=1621x

Exercise 34

34. If f(x)=x2, then f(2+h)=4+h

Answer

False. f(2+h)=(2+h)2=4+4h+h2

Exercise 35

35. The slope of the straight line which is perpendicular to the straight line 2x3y+11=0 is 3/2.

Answer

False. The slope would be 32.

Exercise 36

36. Given that A=(1,5) and B=(7,3), the midpoint of the line segment AB is the point M is given by
M=(172,532)

Answer

False. The midpoint would be M=(1+72,5+32).

Exercise 37

37. (4x2+1)=2x+1

Answer

False. 4x2+1 cannot be simplified any further.

Exercise 38

38. sin(cos(x))=sin(x)cos(x)

Answer

False. sin(cosx) cannot be simplified any further.

Exercise 39

39. x(cos(x)+sin(x))=cos(x2)+sin(x2)=x2cos+x2sin

Answer

False. x(cosx+sinx)=xcosx+xsinx$

Exercise 40

40. 1cos(8x)=2sin2(8x)

Answer

False. 1cos(8x) cannot be simplified any further.

Exercise 41

41. tan(x2)=tanx2

Answer

False. tan(x2) cannot be simplified any further. Furthermore, tan(x2) is the composition of tan(x) with x2. It is not the same as multiplication.

Exercise 42

42. a=a

Answer

False. a=a1/4

Exercise 43

43. 382=(38)2

Answer

True.

Exercise 44

44. 279=9=3

Answer

False. 279=339=33

Exercise 45

45. a21=a

Answer

False. This is an equation that can be solved for a but it is not true in general.

Exercise 46

46. a3/4=1a4/3

Answer

False. a3/4=1a3/4

Exercise 47

47. log2x=2logx

Answer

False. log22(x)=(log2x)2

Exercise 48

48. cos3x=cosx3

Answer

False. cos3(x)=(cosx)3

Exercise 49

49. Recall that log1=0. Then, log(1)=log1=0.

Answer

False. loga(b) where b is a negative number is always undefined. Therefore, \log (-1) is undefined.

Exercise \PageIndex{50}

50. (x-1)^2-(y+1)^2=4 represents a circle centered at (1,-1) and having radius 2.

Answer

False. The equation represents a hyperbola centered at (1,-1).

Exercise \PageIndex{51}

51. y^2-3x+1=0 is an equation of a straight line with slope 3.

Answer

False. y-3x+1=0 is an equation of a straight line with slope equal to 3.

Exercise \PageIndex{52}

52. In the triangle given \sin \alpha =\displaystyle \frac{a}{c}

ex952.jpg

Answer

False. \sin \alpha is defined for right angled triangles.

Exercise \PageIndex{53}

53. The area of the triangle is 1/2 ab.

ex953.jpg

Answer

False. The area of a triangle is \displaystyle{\displaystyle \frac{1}{2} bh}, where b is the base and h is the height of the triangle.

Exercise \PageIndex{54}

54. Two triangles are said to be similar if and only if they have the same area.

Answer

False. Two triangles are said to be similar if and only if their angles are equal.

Exercise \PageIndex{55}

55. Two triangles are said to be similar if and only if they have sides of the same length.

Answer

False. See question above.

Exercise \PageIndex{56}

56. Two triangles are said to be similar if and only if their angles are equal.

Answer

True.

Exercise \PageIndex{57}

57. An isoceles triangle is a triangle in which all the sides have the same length.

Answer

False. An isosceles triangle is a triangle in which two sides have the same length.

Exercise \PageIndex{58}

58. An equilateral triangle must also be right angled.

Answer

False. The angles of equilateral triangle are all 60^{\circ}.

Exercise \PageIndex{59}

59. The centre of the circle (x-1)^2+(y+3)^2=5 is c=(-1,3).

Answer

False. The centre of the circle is (1,-3).

Exercise \PageIndex{60}

60. To solve the inequality, \displaystyle \frac{x-1}{x} >0, we proceed as follows:

Multiply both sides by x to get x-1>0 \Rightarrow x>1.

Answer

False. To solve use a sign chart. The process described does not work because we don't know if x is positive or negative. If x is negative, then multiplication by x causes a change in the inequality.

Exercise \PageIndex{61}

61. \sin{\pi}=180^{\circ}

Answer

False. \sin (\pi) = 0

Exercise \PageIndex{62}

62. If x=1 then \sin x=\sin 1=0.17452406.

Answer

False. \sin (1) = 0.841470985

Exercise \PageIndex{63}

63. If \sin x= \displaystyle \frac{1}{2} then \sin x= 30^{\circ}.

Answer

True.

Exercise \PageIndex{64}

64. \cos(-x)= \cos x

Answer

False. \cos (-x) = \cos (x)

Exercise \PageIndex{65}

65. On the number line shown below, the collection of real numbers represented is [ - \infty,2).

ex965.jpg

Answer

False. The collection is (-\infty, 2).

Exercise \PageIndex{66}

66. On the number line shown above, the collection of real numbers represented is (2, - \infty).

Answer

False. See question above.

Exercise \PageIndex{67}

67. The interval (-5,4] can be represented on the real number line as shown below.

ex957.jpg

Answer

False. The interval is [-5,4).

Exercise \PageIndex{68}

68. \sqrt[4]{x^3}=x^{4/3}

Answer

False. \sqrt[4]{x^3} = x^{3/4}

Exercise \PageIndex{69}

69. \sqrt{64}= \pm 8

Answer

False. \sqrt{64} = 8

Exercise \PageIndex{70}

70. \sqrt{-1}= -1

Answer

False. \sqrt{-1} = i

Exercise \PageIndex{71}

71. \sqrt[3]{x^2}= x^{2/3}

Answer

True.

Exercise \PageIndex{72}

72. The pair of straight lines 2x+y-1=0 and 2y+x-1=0 are parallel.

Answer

False. The lines are neither parallel nor perpendicular.

Exercise \PageIndex{73}

73. The point of intersection of the parallel lines 3x+2y-1=0, and 6x+4y+17=0 is (-1,4).

Answer

False. There is no point of intersection of parallel lines.

Exercise \PageIndex{74}

74. The graphs of y=-2 and x=5 are as shown below.

ex741.jpg

ex742.jpg

Answer

False. The graphs of x = -2 and y = 5 are shown.

Exercise \PageIndex{75}

75. The straight line 2x+3y=12 has an x-intercept of 6 and y-intercept of 4. Therefore, the line must contain the point (6,4).

Answer

False. The intercepts are correct but the line would contain the points (6,0) and (0,4).

Exercise \PageIndex{76}

76. If \sin x= \displaystyle \frac{3}{5}, then \sec x= \displaystyle \frac{5}{3}.

Answer

False. If \sin (x) = 3/5, then \csc (x) = 5/3.

Exercise \PageIndex{77}

77. |x^2|=-x^2 if x<0.

Answer

False. |x^2| = x^2 for all x.

Exercise \PageIndex{78}

78. |-x|=x if x<0.

Answer

False. |-x| = x if x>0.

Exercise \PageIndex{79}

79. |-3x|=-3x if x<0.

Answer

True.

Exercise \PageIndex{80}

80. x^2y=xy^2.

Answer

False. There is no way of fixing this.

Exercise \PageIndex{81}

81. \sqrt[3]{-x} is not defined for any real number values of x.

Answer

False. \sqrt[3]{-x} is defined for all real numbers.

Exercise \PageIndex{82}

82. If x^3=2, then x=2^3=8.

Answer

False. If x^3 = 2, then x = \sqrt[3]{2}.

Exercise \PageIndex{83}

83. If \sqrt[3]{x}=8, then x=\sqrt[3]{8}=2.

Answer

False. If \sqrt[3]{x} = 8, then x = 8^3=512.

Exercise \PageIndex{84}

84. The equation \sqrt[3]{x}=-1 has no real number solutions.

Answer

False. The solution is x = -1.

Exercise \PageIndex{85}

85. The graph of y= \cos x is the one shown below:

ex985.jpg

Answer

False. The graph of y = \sin (x) is shown.

Exercise \PageIndex{86}

86. The inequality -1 \leq x \geq 2 is valid.

Answer

True. It is redundant as it is equivalent to x \geq -1.

Exercise \PageIndex{87}

87. The inequality -1 \geq x \geq 2 is valid.

Answer

True.

Exercise \PageIndex{88}

88. The radius of the circle 4x^2+4y^2=81 is equal to 9 units.

Answer

False. The radius is 9/2.

Exercise \PageIndex{89}

89. Every pair of straight lines is either parallel or perpendicular.

Answer

False. For example, y = x and y = 2x are neither parallel nor perpendicular.

Exercise \PageIndex{90}

90. The equation of the x-axis is x=0.

Answer

False. The equation of the x-axis is y=0.

Exercise \PageIndex{91}

91. The equation of the y-axis is x=0.

Answer

True.

Exercise \PageIndex{92}

92. x^2+1=(x+1)(x+1).

Answer

False. x^2+1 is not factorable.

Exercise \PageIndex{93}

93. x^3+1=(x^2+1)(x+1).

Answer

False. x^3+1 = (x+1)(x^2-x+1)

Exercise \PageIndex{94}

94. x^4-1=(x-1)(x-1)(x-1)(x-1).

Answer

False. x^4-1 = (x-1)(x+1)(x^2+1)

Exercise \PageIndex{95}

95. The slope of the straight line 3y=5x-2 is 5.

Answer

False. The slope is 5/3.

Exercise \PageIndex{96}

96. The distance between the points (0,0) and (5,5) is 5 units.

Answer

False. The distance is 5\sqrt{2} units.

Exercise \PageIndex{97}

97. The distance between the points (-5,0) and (5,5) is 0 units.

Answer

False. The distance is 10 units.

Exercise \PageIndex{98}

98. The point-slope formula for the equation of a straight line may be expressed as x-x_1=m(y-y_1) where m is the slope of the straight line and the line contains the point (x_1,y_1).

Answer

False. The equation is y-y_1 - m(x-x_1).

Exercise \PageIndex{99}

99. x^2-10x-24=(x-4)(x-6)

Answer

False. x^2-10x+24 = (x-4)(x-6)

Exercise \PageIndex{100}

100. x^2-10x+24=(x-12)(x+2)

Answer

False. x^2-10x-24 = (x-12)(x+2)

Exercise \PageIndex{101}

101. If ax^2+bx+c=0 then

a. \displaystyle x=\displaystyle \frac {\pm \sqrt{(b^2-4ac)}}{2a}

b. \displaystyle x=\displaystyle \frac {-b \pm \sqrt{(4ac-b^2)}}{2a}

Answer

Both are false. \displaystyle{x = \displaystyle \frac{-b\pm\sqrt{b^2-4ac}}{2a}}

Exercise \PageIndex{102}

102. Question: Simplify \displaystyle \frac{x+1}{x-1}=\frac{x+2}{x-4}

Answer: by cross multiplying we get,

\begin{eqnarray*} (x+1)(x-4) &=& (x+2)(x-1)\\ \Rightarrow x^2-3x-4 &=& x^2+x-2\\ \Rightarrow -4x &=& 2\\ \Rightarrow x=\displaystyle \frac{-1}{2} \end{eqnarray*}

Answer

False. We are asked to simplify an expression not an equation. Therefore are solution should be a fraction and not x = something. The answer is obtained by making the expression into a single expression: \displaystyle{\displaystyle \frac{(x+1)(x+2)}{(x-1)(x-4)}}.

Exercise \PageIndex{103}

103. If \displaystyle \frac{x+1}{x-1}=0 then (x-1)=0(x^2+3)\Rightarrow x-1=x^2+3

Answer

False. If we have x-1 = 0 \cdot (x^2+3), then x-1 = 0.

Exercise \PageIndex{104}

104. If \log_2x=5 then one of the following is true:

a. x=5^2

b. x^2=5

c. 5^x=2

d. x=2^5

e. 2^x=5

f. None of the above.

Answer

d. is true.

Exercise \PageIndex{105}

105. If f(x)=\displaystyle \frac{x^2-4}{x-2} then f(2)=\displaystyle \frac{4-4}{2-2}=\displaystyle \frac{0}{0}=1.

Answer

False. f(2) is undefined because of division by zero.

Exercise \PageIndex{106}

106. \displaystyle \frac{2}{x}+\displaystyle \frac{5}{y}=\displaystyle \frac{2y+5x}{x+y}

Answer

True.

Exercise \PageIndex{107}

107. To sketch a straight line, you need at least three points which lie on the straight line.

Answer

False. To sketch a straight line you need two points.

Exercise \PageIndex{108}

108. A circle is completely determined if its centre and radius are known.

Answer

True.

Exercise \PageIndex{109}

109. \sqrt{-x} is undefined because of the negative sign.

Answer

False. \sqrt{-x} is defined if x<0.

Exercise \PageIndex{110}

110. \log (xy)= (\log x) (\log y)

Answer

False. \log(xy) = \log (x) + \log (y)

Exercise \PageIndex{111}

111. 3^0=0

Answer

False. 3^0 = 1


This page titled Common Mistakes is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Pamini Thangarajah.

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