Equazioni di secondo grado

Algebra

Equazioni di secondo grado

equazioni monomie

1 𝑥𝑥² = 0 3𝑥𝑥² = 0

2 5

7 𝑥𝑥² = 0 2

3 𝑥𝑥² = 0

3 4𝑥𝑥² = 0 10𝑥𝑥² = 0

equazioni pure

4 𝑥𝑥²− 1 = 0 ^𝑥𝑥1= −1, 𝑥𝑥2= 1 𝑥𝑥² − 4 = 0 ^𝑥𝑥1= −2, 𝑥𝑥2= 2

5 3𝑥𝑥²− 27 = 0 ^𝑥𝑥1= −3, 𝑥𝑥2= 3 9𝑥𝑥² − 4 = 0 𝑥𝑥1= −2 3 , 𝑥𝑥²=2

3

6 𝑥𝑥²+ 1 = 0 ^{∄ 𝑥𝑥 ∈ 𝑅𝑅} 𝑥𝑥² + 9 = 0 ^{∄ 𝑥𝑥 ∈ 𝑅𝑅}

7 3𝑥𝑥²− 1 = 0 𝑥𝑥₁= −√3

3 , 𝑥𝑥²=√3

3 5𝑥𝑥² + 125 = 0 ^{∄𝑥𝑥 ∈ 𝑅𝑅}

8 125𝑥𝑥²− 5 = 0 𝑥𝑥1= −1

5 , 𝑥𝑥²=1

5 4 + 124𝑥𝑥² = 0 ^{∄ 𝑥𝑥 ∈ 𝑅𝑅}

9 𝑥𝑥²− √2 = 0 ^{𝑥𝑥1= −√24 , 𝑥𝑥2= √24} √3𝑥𝑥²+ 1 = 0 ^{∄ 𝑥𝑥 ∈ 𝑅𝑅}

10 7

8 𝑥𝑥²− 1 = 0 ^𝑥𝑥1= −2√14

7 , 𝑥𝑥²=2√14 7

4

5 𝑥𝑥² − 49 = 0 𝑥𝑥₁= −7√5

2 , 𝑥𝑥²=7√5 2

11 7 + 49𝑥𝑥² = 0 ^{∄ 𝑥𝑥 ∈ 𝑅𝑅} 7 − 49𝑥𝑥² = 0 𝑥𝑥₁= −√7

7 , 𝑥𝑥²=√7 7

12 5𝑥𝑥² = 1 𝑥𝑥1= −√5

5 , 𝑥𝑥²=√5

5 7 = −3𝑥𝑥^{2 ∄ 𝑥𝑥 ∈ 𝑅𝑅}

equazioni spurie

13 𝑥𝑥²− 𝑥𝑥 = 0 ^𝑥𝑥1= 0, 𝑥𝑥₂= 1 3𝑥𝑥² + 5𝑥𝑥 = 0 𝑥𝑥1= 0, 𝑥𝑥2= −5 3

14 7

9 𝑥𝑥²− 𝑥𝑥 = 0 ^𝑥𝑥1= 0, 𝑥𝑥₂=9

7 𝑥𝑥² + 4𝑥𝑥 = 0 ^{𝑥𝑥1= 0, x2= −4}

15 𝑥𝑥²+2

3 𝑥𝑥 = 0 ^{𝑥𝑥1= 0, 𝑥𝑥2= −}

2

3 5𝑥𝑥² + 10𝑥𝑥 = 0 ^𝑥𝑥1= 0, 𝑥𝑥2= −2

16 81

4 𝑥𝑥²−9

2 𝑥𝑥 = 0 ^{𝑥𝑥1= 0, 𝑥𝑥2=}

2 9

9

4 𝑥𝑥² −2

3 𝑥𝑥 = 0 ^{𝑥𝑥1= 0, 𝑥𝑥2=}

8 27 17 𝑥𝑥 − 2𝑥𝑥² = 0 𝑥𝑥1= 0, x2=1

2 2𝑥𝑥 + 𝑥𝑥² = 0 ^𝑥𝑥1= 0, 𝑥𝑥₂= −2

18 3

5 𝑥𝑥²+ 𝑥𝑥 = 0 ^𝑥𝑥1= 0, 𝑥𝑥₂= −5 3

25

49 𝑥𝑥² + 𝑥𝑥 = 0 ^𝑥𝑥1= 0, 𝑥𝑥₂= −49 25 19 5𝑥𝑥 = 5𝑥𝑥^{2 𝑥𝑥}1= 0, x2= 1 5𝑥𝑥 − 3𝑥𝑥² = 0 𝑥𝑥₁= 0, 𝑥𝑥₂=5

3

Algebra

Equazioni di secondo grado

20 𝑥𝑥² = √5𝑥𝑥 ^{𝑥𝑥1= 0, 𝑥𝑥2= √5} √2𝑥𝑥² = −2𝑥𝑥 ^{𝑥𝑥1= 0, 𝑥𝑥2= −√2}

equazioni complete

21 𝑥𝑥²− 7𝑥𝑥 + 10 = 0 ^𝑥𝑥1= 2, 𝑥𝑥₂= 5

22 𝑥𝑥²+ 5𝑥𝑥 + 6 = 0 ^𝑥𝑥1= −3, 𝑥𝑥2= −2

23 𝑥𝑥²+ 2𝑥𝑥 − 3 = 0 ^{𝑥𝑥1= −3, 𝑥𝑥2= 1}

24 2𝑥𝑥²+ 7𝑥𝑥 + 3 = 0 𝑥𝑥1= −3, 𝑥𝑥2= −1

2

25 𝑥𝑥²+ 𝑥𝑥 + 12 = 0 ^{∄ 𝑥𝑥 ∈ 𝑅𝑅}

26 5𝑥𝑥²− 2x − 16 = 0 𝑥𝑥₁= −8

5 , 𝑥𝑥²= 2

27 2𝑥𝑥²+ 5𝑥𝑥 − 12 = 0 𝑥𝑥1= −4, 𝑥𝑥2=3

2

28 25𝑥𝑥²− 20𝑥𝑥 + 4 = 0 𝑥𝑥1= 𝑥𝑥2=2

5

29 𝑥𝑥²− 5𝑥𝑥 + 6 = 0 ^𝑥𝑥1= 2, 𝑥𝑥2= 3

30 𝑥𝑥²+ 𝑥𝑥 − 12 = 0 ^{𝑥𝑥1= −4, 𝑥𝑥2= 3}

31 𝑥𝑥²+ 7𝑥𝑥 − 8 = 0 ^𝑥𝑥1= −8, 𝑥𝑥₂= 1

32 9𝑥𝑥²− 6𝑥𝑥 + 1 = 0 𝑥𝑥1= 𝑥𝑥2=1

3

33 𝑥𝑥²− 𝑥𝑥 = 20 ^{𝑥𝑥1= −4, 𝑥𝑥2= 5}

34 3𝑥𝑥²+ 5𝑥𝑥 + 3 = 0 ^{∄ 𝑥𝑥 ∈ 𝑅𝑅}

35 2𝑥𝑥² = 3𝑥𝑥 + 2 𝑥𝑥1= −1

2 , 𝑥𝑥²= 2

36 𝑥𝑥² = 𝑥𝑥 + 2 ^𝑥𝑥1= −1, 𝑥𝑥2= 2

Algebra

Equazioni di secondo grado

37 10 = 3𝑥𝑥² + 𝑥𝑥 ^𝑥𝑥1= −2, 𝑥𝑥₂=5

3

38 𝑥𝑥²− 3√2𝑥𝑥 + 4 = 0 ^𝑥𝑥1= √2, 𝑥𝑥2= 2√2

39 3𝑥𝑥²− 𝑥𝑥 + 1 = 0 ^{∄ 𝑥𝑥 ∈ 𝑅𝑅}

40 4𝑥𝑥²− 4𝑥𝑥 − 1 = 0 _𝑥𝑥1=^{1 − √2}

2 , 𝑥𝑥2=1 + √2 2

41 𝑥𝑥²+ 4𝑥𝑥 + 1 = 0 ^𝑥𝑥1= −2 − √3, 𝑥𝑥₂= −2 + √3

42 𝑥𝑥² = −𝑥𝑥 − 2 ^{∄ 𝑥𝑥 ∈ 𝑅𝑅}

43 𝑥𝑥²− �√3 + √2�𝑥𝑥 + √6 = 0 ^𝑥𝑥1= √2, 𝑥𝑥2= √3

44 (2𝑥𝑥 + 1)²− (𝑥𝑥 − 3)² = 0 ^𝑥𝑥1= −4, 𝑥𝑥₂=2

3

45 𝑥𝑥 = (𝑥𝑥 + 2)^{2 ∄ 𝑥𝑥 ∈ 𝑅𝑅}

46 𝑥𝑥²+ �√2 − √3�𝑥𝑥 − √6 = 0 ^𝑥𝑥1= −√2, 𝑥𝑥2= √3

47 𝑥𝑥 3

2+ 3𝑥𝑥1 + 3 = 0 𝑥𝑥₁=−9 − 3√5

2 , 𝑥𝑥₂=−9 + 3√5 2 48 𝑥𝑥²−1

2 + 1 6 𝑥𝑥 = −

1

4 ^{𝑥𝑥1=−1 − √3712 , 𝑥𝑥2=−1 + √3712}

49 𝑥𝑥²+ 16 = 6√2𝑥𝑥 ^{𝑥𝑥1= 2√2, 𝑥𝑥2= 4√2}

50 4√5𝑥𝑥 − 1 = 4𝑥𝑥² 𝑥𝑥₁=√5 − 2

2 , 𝑥𝑥₂=√5 + 2 2

51 𝑥𝑥²− 4√3𝑥𝑥 + 12 = 0 ^𝑥𝑥1= 𝑥𝑥₂= 2√3

52 𝑥𝑥 + √5 = �2𝑥𝑥 − √5��𝑥𝑥 + √5� 𝑥𝑥1= −√5, 𝑥𝑥2=1 + √5 2 53 �𝑥𝑥 +1

2�

2

= �3 2 − 2𝑥𝑥�

2

𝑥𝑥1= 2, 𝑥𝑥2=1 3

54 𝑥𝑥�𝑥𝑥 + √7� = 14 ^𝑥𝑥1= −2√7, 𝑥𝑥₂= √7

Algebra

Equazioni di secondo grado

55 1

3(3 − 𝑥𝑥) +𝑥𝑥²− 1 6 −

2

3 = 0 ^{𝑥𝑥1= 𝑥𝑥2= 1}

56 2√3𝑥𝑥 = 𝑥𝑥² + 2 ^𝑥𝑥1= √3 − 1, 𝑥𝑥2= √3 + 1

equazioni di secondo grado frazionarie

57 2

𝑥𝑥 = 𝑥𝑥

2 ^{𝑥𝑥1= −2, 𝑥𝑥2= 2}

58 𝑥𝑥 =1

𝑥𝑥 ^{𝑥𝑥1= −1, 𝑥𝑥2= 1}

59 √3

𝑥𝑥 = 𝑥𝑥

√27 ^{𝑥𝑥1= −3, 𝑥𝑥2= 3}

60 𝑥𝑥 − 1 = 9

𝑥𝑥 − 1 ^{𝑥𝑥1= −2, 𝑥𝑥2= 4}

61 − 2

𝑥𝑥 + 3 = 1

𝑥𝑥^{2 ∄ 𝑥𝑥 ∈ 𝑅𝑅}

62 1

𝑥𝑥² = 2

𝑥𝑥 + 3 ^{𝑥𝑥1= −1, 𝑥𝑥2=}

3 2

63 2

𝑥𝑥²+ 1 =

1 − 𝑥𝑥²

𝑥𝑥²+ 1 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖

64 3

𝑥𝑥 − 1 = 2𝑥𝑥

𝑥𝑥 + 1 ^{𝑥𝑥1= −}

1

2 , 𝑥𝑥²= 3

65 3

𝑥𝑥(𝑥𝑥 − 2) =

4𝑥𝑥 − 2

𝑥𝑥 ^{𝑥𝑥1=5 − √21}4 , 𝑥𝑥₂=5 + √21

4

66 4

3 + 1 𝑥𝑥 − 1 =

1 + 𝑥𝑥 𝑥𝑥 − 1 −

𝑥𝑥

𝑥𝑥 + 1 ^{𝑥𝑥1= −}

1

2 , 𝑥𝑥²= 2

67 1

𝑥𝑥² = 2 − 𝑥𝑥

𝑥𝑥 ^{𝑥𝑥1= 𝑥𝑥2= 1}

68 𝑥𝑥 + 3 𝑥𝑥 −

1

𝑥𝑥² = −1

𝑥𝑥 ^𝑥𝑥1= −2 − √5 , 𝑥𝑥₂= −2 + √5

69 3𝑥𝑥

𝑥𝑥 + 1 +

𝑥𝑥²

2𝑥𝑥²+ 𝑥𝑥 − 1 = 2𝑥𝑥

1 − 2𝑥𝑥 ^{𝑥𝑥1= 0, 𝑥𝑥2=}

1 9 70 11 − 3𝑥𝑥

4(𝑥𝑥 − 1) = 1 + 𝑥𝑥 𝑥𝑥²− 1 −

1

𝑥𝑥 + 1 ^{𝑥𝑥1= −13 , 𝑥𝑥2= 3}

71 1

𝑥𝑥²− 1 + 𝑥𝑥 + 1 𝑥𝑥 − 1 =

1 − 𝑥𝑥

𝑥𝑥 + 1 ^{∄ 𝑥𝑥 ∈ 𝑅𝑅}

72 2

𝑥𝑥 + 1 + 𝑥𝑥 1 − 𝑥𝑥 =

8

1 − 𝑥𝑥^{2 𝑥𝑥1= −2, 𝑥𝑥2= 3}

Algebra

Equazioni di secondo grado

73 3𝑥𝑥 + 2 𝑥𝑥²+ 2𝑥𝑥 +

1 𝑥𝑥 =

𝑥𝑥 + 1

𝑥𝑥 + 2 ^{𝑥𝑥1= −1, 𝑥𝑥2= 4}

74 2

𝑥𝑥 − 2𝑥𝑥 𝑥𝑥 + 1 =

2 − 𝑥𝑥

𝑥𝑥 ^{𝑥𝑥1= 1, 𝑥𝑥2}= 0 𝑖𝑖𝑛𝑛𝑖𝑖 𝑖𝑖𝑎𝑎𝑎𝑎𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑎𝑎𝑖𝑖𝑎𝑎𝑖𝑖

75 𝑥𝑥²− 11

𝑥𝑥²+ 2𝑥𝑥 − 3 + 𝑥𝑥 𝑥𝑥 − 1 =

𝑥𝑥 + 1

𝑥𝑥 + 3 ^{𝑥𝑥1= −5, 𝑥𝑥2= 2}

equazioni di secondo grado di vario tipo

76 4𝑥𝑥(2 − 𝑥𝑥) + (𝑥𝑥 − 2) = −11 ^𝑥𝑥1= −3, 𝑥𝑥2= 3

77 �√3 − 2𝑥𝑥�² = �𝑥𝑥 + √3�^{2 𝑥𝑥1= 0, 𝑥𝑥2= 2√3}

78 𝑥𝑥�𝑥𝑥 + √7� = 14 𝑥𝑥1= −2√7, 𝑥𝑥2= √7

79 4 + (1 + 2𝑥𝑥)² = 0 ^{∄ 𝑥𝑥 ∈ 𝑅𝑅}

80 1

2(𝑥𝑥 − 1) + 𝑥𝑥 = (𝑥𝑥 + 1)(𝑥𝑥 − 2) ^𝑥𝑥1= −1

2 , 𝑥𝑥²= 3 81 (2𝑥𝑥 − 1)²+ 4(𝑥𝑥 − 2)(𝑥𝑥 + 2) = −18 ^{∄ 𝑥𝑥 ∈ 𝑅𝑅}

82 1

3(3 − 𝑥𝑥) +𝑥𝑥²− 1 6 −

2

3 = 0 ^{𝑥𝑥1= 𝑥𝑥2= 1}

83 √2(2𝑥𝑥 + 1) + �𝑥𝑥 − √2�² = 𝑥𝑥 + 4 ^𝑥𝑥1= 1 − √2, 𝑥𝑥₂= √2

84 2√3𝑥𝑥 = 𝑥𝑥² + 2 ^𝑥𝑥1= √3 − 1, 𝑥𝑥2= √3 + 1

85 1

𝑥𝑥 + 1 + 1 12 =

1

1 − 𝑥𝑥 ^𝑥𝑥1= −12 − √145, 𝑥𝑥2= −12 + √145

86 1

𝑥𝑥 + √3+ 1

𝑥𝑥 − √3= 3 _𝑥𝑥1=^{1 − 2√7}

3 , 𝑥𝑥2=1 + 2√7 3 87 𝑥𝑥 + 2 = 6

𝑥𝑥 + 2 + 1 ^{𝑥𝑥1= −4, 𝑥𝑥2= 1}

88 𝑥𝑥 − 5 8 −

𝑥𝑥 5 =

5

𝑥𝑥 − 10 ^{𝑥𝑥1= −}

10

3 , 𝑥𝑥2= 5

89 1

𝑥𝑥(2 − 𝑥𝑥) −

4 − 𝑥𝑥 𝑥𝑥(𝑥𝑥 + 2) =

2

4 − 𝑥𝑥^{2 𝑥𝑥1}= 2 𝑖𝑖𝑛𝑛𝑖𝑖 𝑖𝑖𝑎𝑎𝑎𝑎𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑎𝑎𝑖𝑖𝑎𝑎𝑖𝑖, 𝑥𝑥2= 3

90 1

𝑥𝑥(𝑥𝑥 + 3) − 1 𝑥𝑥(2 − 𝑥𝑥) =

1

6𝑥𝑥 ^{𝑥𝑥1= −1, 𝑥𝑥2= 12}

Algebra

Equazioni di secondo grado

91 𝑥𝑥 + 2 𝑥𝑥 −

(2 + 3𝑥𝑥)

2(4 − 3𝑥𝑥) = 1 + 4

3𝑥𝑥² − 4𝑥𝑥 ^{𝑥𝑥1= −6, 𝑥𝑥2=}

4

3 𝑖𝑖𝑛𝑛𝑖𝑖 𝑖𝑖𝑎𝑎𝑎𝑎𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑎𝑎𝑖𝑖𝑎𝑎𝑖𝑖

92 5

4𝑥𝑥 − 9 4 =

𝑥𝑥 − 3

𝑥𝑥 + 1 ^{𝑥𝑥1= −}

5

13 , 𝑥𝑥₂= 1

93 𝑥𝑥

𝑥𝑥 − 6 − 1 2 =

𝑥𝑥 6 +

𝑥𝑥 + 6

6 − 𝑥𝑥 ^{𝑥𝑥1= −3, 𝑥𝑥2= 18}

94 4

(𝑥𝑥 − 2)²+ 3 = 3𝑥𝑥

2 − 𝑥𝑥 ^{∄ 𝑥𝑥 ∈ 𝑅𝑅}

95 1

3𝑥𝑥 − 𝑥𝑥²− 1 6𝑥𝑥 =

1

2𝑥𝑥 − 𝑥𝑥^{2 ∄ 𝑥𝑥 ∈ 𝑅𝑅}

96 𝑥𝑥 − √3

𝑥𝑥√3 − 1= 2√3 3𝑥𝑥²− 1 −

𝑥𝑥

𝑥𝑥√3 + 1 ^{𝑥𝑥1= −}

√3

2 , 𝑥𝑥²= √3

97 4

(𝑥𝑥 − 2)²− 3

𝑥𝑥² = 5

4𝑥𝑥 − 2𝑥𝑥^{2 𝑥𝑥1=−7 − √217}7 , 𝑥𝑥2=−7 + √217

7

98 5

2𝑥𝑥 + 2 + 1 2𝑥𝑥 − 2 =

3𝑥𝑥 + 2

𝑥𝑥²+ 𝑥𝑥 + 1 ^{𝑥𝑥1= 0, 𝑥𝑥2= 4}

99 5

6 + 2𝑥𝑥 − 1 𝑥𝑥²− 3𝑥𝑥 =

5

9 − 𝑥𝑥^{2 𝑥𝑥1= −}

3

5 , 𝑥𝑥2= 2

100 2

(𝑥𝑥 − 1)²− �𝑥𝑥 − 1

𝑥𝑥 − 1� �𝑥𝑥 − 1

𝑥𝑥 + 1� + �𝑥𝑥 − 1 𝑥𝑥 − 1�

2

= 0 ^𝑥𝑥1= 1 − √3, 𝑥𝑥₂= 1 + √3

101 1

𝑥𝑥 − 5 − 1 4 − 𝑥𝑥 =

2 𝑥𝑥 − 4 +

1

𝑥𝑥 − 3 ^𝑥𝑥1= 5 − √2, 𝑥𝑥2= 5 + √2

102 𝑥𝑥 + 3 𝑥𝑥 + 1 +

𝑥𝑥 + 5 𝑥𝑥 + 2 =

2𝑥𝑥 + 2

𝑥𝑥 ^{𝑥𝑥1= −}

4

3 , 𝑥𝑥2= 1

103 𝑥𝑥²

200 − 𝑥𝑥² =1

7 ^{𝑥𝑥1= −5, 𝑥𝑥2= 5}

104 2(𝑥𝑥 − 1) 𝑥𝑥 + 1 +

(𝑥𝑥 + 1)

4(𝑥𝑥 − 1) = 3 ^{𝑥𝑥1=−7 − 4√7}3 , 𝑥𝑥₂=−7 + 4√7

3

105 𝑥𝑥 − 2 3 − 𝑥𝑥 +

2𝑥𝑥 + 9 𝑥𝑥²− 9 =

2𝑥𝑥 + 1

𝑥𝑥 + 3 ^𝑥𝑥1= 1 − √7, 𝑥𝑥₂= 1 + √7

106 5𝑥𝑥²− 5𝑥𝑥 − 28 𝑥𝑥²− 𝑥𝑥 − 6 =

4 𝑥𝑥 + 2 −

2

𝑥𝑥 − 3 ^{𝑥𝑥1= −1, 𝑥𝑥2=}

12 5