Algebra
Equazioni di secondo grado
equazioni monomie
1
𝑥𝑥 2 = 0
3𝑥𝑥 2 = 0
4𝑥𝑥 2 = 0
10𝑥𝑥 2 = 0
2 2
𝑥𝑥 = 0
3
5 2
𝑥𝑥 = 0
7
2
3
4
𝑥𝑥 2 − 1 = 0
𝑥𝑥1 = −1, 𝑥𝑥2 = 1
𝑥𝑥 2 + 1 = 0
∄ 𝑥𝑥 ∈ 𝑅𝑅
3𝑥𝑥 2 − 27 = 0
5
6
3𝑥𝑥 2 − 1 = 0
7
125𝑥𝑥 2 − 5 = 0
8
9
10
11
12
13
14
15
16
17
18
19
equazioni pure
𝑥𝑥 2 − √2 = 0
7 2
𝑥𝑥 − 1 = 0
8
7 + 49𝑥𝑥 2 = 0
5𝑥𝑥 2 = 1
𝑥𝑥 2 − 𝑥𝑥 = 0
7 2
𝑥𝑥 − 𝑥𝑥 = 0
9
2
𝑥𝑥 2 + 𝑥𝑥 = 0
3
81 2 9
𝑥𝑥 − 𝑥𝑥 = 0
4
2
𝑥𝑥 − 2𝑥𝑥 2 = 0
3 2
𝑥𝑥 + 𝑥𝑥 = 0
5
5𝑥𝑥 = 5𝑥𝑥 2
v 3.0
√3
√3
, 𝑥𝑥 =
3 2
3
1
1
𝑥𝑥1 = − , 𝑥𝑥2 =
5
5
4
4
𝑥𝑥1 = − √2, 𝑥𝑥2 = √2
𝑥𝑥1 = −
2√14
2√14
, 𝑥𝑥2 =
7
7
∄ 𝑥𝑥 ∈ 𝑅𝑅
𝑥𝑥1 = −
𝑥𝑥1 = −2, 𝑥𝑥2 = 2
𝑥𝑥 2 + 9 = 0
∄ 𝑥𝑥 ∈ 𝑅𝑅
9𝑥𝑥 2 − 4 = 0
𝑥𝑥1 = −3, 𝑥𝑥2 = 3
𝑥𝑥1 = −
𝑥𝑥 2 − 4 = 0
√5
√5
, 𝑥𝑥2 =
5
5
5𝑥𝑥 2 + 125 = 0
4 + 124𝑥𝑥 2 = 0
√3𝑥𝑥 2 + 1 = 0
4 2
𝑥𝑥 − 49 = 0
5
7 − 49𝑥𝑥 2 = 0
7 = −3𝑥𝑥 2
2
2
𝑥𝑥1 = − , 𝑥𝑥2 =
3
3
∄𝑥𝑥 ∈ 𝑅𝑅
∄ 𝑥𝑥 ∈ 𝑅𝑅
∄ 𝑥𝑥 ∈ 𝑅𝑅
𝑥𝑥1 = −
𝑥𝑥1 = −
7√5
7√5
, 𝑥𝑥2 =
2
2
√7
√7
, 𝑥𝑥2 =
7
7
∄ 𝑥𝑥 ∈ 𝑅𝑅
equazioni spurie
𝑥𝑥1 = 0, 𝑥𝑥2 = 1
𝑥𝑥1 = 0, 𝑥𝑥2 =
9
7
𝑥𝑥1 = 0, 𝑥𝑥2 = −
𝑥𝑥1 = 0, 𝑥𝑥2 =
𝑥𝑥1 = 0, x2 =
2
9
1
2
𝑥𝑥1 = 0, 𝑥𝑥2 = −
𝑥𝑥1 = 0, x2 = 1
2
3
5
3
5
3
3𝑥𝑥 2 + 5𝑥𝑥 = 0
𝑥𝑥1 = 0, 𝑥𝑥2 = −
5𝑥𝑥 2 + 10𝑥𝑥 = 0
𝑥𝑥1 = 0, 𝑥𝑥2 = −2
𝑥𝑥 2 + 4𝑥𝑥 = 0
9 2 2
𝑥𝑥 − 𝑥𝑥 = 0
4
3
2𝑥𝑥 + 𝑥𝑥 2 = 0
25 2
𝑥𝑥 + 𝑥𝑥 = 0
49
5𝑥𝑥 − 3𝑥𝑥 2 = 0
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𝑥𝑥1 = 0, x2 = −4
𝑥𝑥1 = 0, 𝑥𝑥2 =
8
27
𝑥𝑥1 = 0, 𝑥𝑥2 = −2
𝑥𝑥1 = 0, 𝑥𝑥2 = −
𝑥𝑥1 = 0, 𝑥𝑥2 =
5
3
49
25
1 di 6
Algebra
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𝑥𝑥 2 = √5𝑥𝑥
Equazioni di secondo grado
𝑥𝑥1 = 0, 𝑥𝑥2 = √5
√2𝑥𝑥 2 = −2𝑥𝑥
𝑥𝑥1 = 0, 𝑥𝑥2 = −√2
equazioni complete
𝑥𝑥 2 − 7𝑥𝑥 + 10 = 0
𝑥𝑥1 = 2, 𝑥𝑥2 = 5
𝑥𝑥 2 + 5𝑥𝑥 + 6 = 0
𝑥𝑥1 = −3, 𝑥𝑥2 = −2
𝑥𝑥 2 + 2𝑥𝑥 − 3 = 0
𝑥𝑥1 = −3, 𝑥𝑥2 = 1
2𝑥𝑥 2 + 7𝑥𝑥 + 3 = 0
𝑥𝑥1 = −3, 𝑥𝑥2 = −
𝑥𝑥 2 + 𝑥𝑥 + 12 = 0
∄ 𝑥𝑥 ∈ 𝑅𝑅
2𝑥𝑥 2 + 5𝑥𝑥 − 12 = 0
𝑥𝑥1 = −4, 𝑥𝑥2 =
8
𝑥𝑥1 = − , 𝑥𝑥2 = 2
5
5𝑥𝑥 2 − 2x − 16 = 0
25𝑥𝑥 2 − 20𝑥𝑥 + 4 = 0
𝑥𝑥1 = 𝑥𝑥2 =
𝑥𝑥 2 − 5𝑥𝑥 + 6 = 0
2
5
3
2
𝑥𝑥1 = 2, 𝑥𝑥2 = 3
𝑥𝑥 2 + 𝑥𝑥 − 12 = 0
𝑥𝑥1 = −4, 𝑥𝑥2 = 3
𝑥𝑥 2 + 7𝑥𝑥 − 8 = 0
𝑥𝑥1 = −8, 𝑥𝑥2 = 1
9𝑥𝑥 2 − 6𝑥𝑥 + 1 = 0
𝑥𝑥1 = 𝑥𝑥2 =
𝑥𝑥 2 − 𝑥𝑥 = 20
1
3
𝑥𝑥1 = −4, 𝑥𝑥2 = 5
3𝑥𝑥 2 + 5𝑥𝑥 + 3 = 0
∄ 𝑥𝑥 ∈ 𝑅𝑅
𝑥𝑥 2 = 𝑥𝑥 + 2
𝑥𝑥1 = −1, 𝑥𝑥2 = 2
1
𝑥𝑥1 = − , 𝑥𝑥2 = 2
2
2𝑥𝑥 2 = 3𝑥𝑥 + 2
v 3.0
1
2
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2 di 6
Equazioni di secondo grado
Algebra
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54
10 = 3𝑥𝑥 2 + 𝑥𝑥
𝑥𝑥1 = −2, 𝑥𝑥2 =
5
3
𝑥𝑥 2 − 3√2𝑥𝑥 + 4 = 0
𝑥𝑥1 = √2, 𝑥𝑥2 = 2√2
4𝑥𝑥 2 − 4𝑥𝑥 − 1 = 0
𝑥𝑥1 =
3𝑥𝑥 2 − 𝑥𝑥 + 1 = 0
∄ 𝑥𝑥 ∈ 𝑅𝑅
𝑥𝑥 2 + 4𝑥𝑥 + 1 = 0
𝑥𝑥1 = −2 − √3, 𝑥𝑥2 = −2 + √3
𝑥𝑥 2 = −𝑥𝑥 − 2
∄ 𝑥𝑥 ∈ 𝑅𝑅
𝑥𝑥 2 − �√3 + √2�𝑥𝑥 + √6 = 0
𝑥𝑥1 = √2, 𝑥𝑥2 = √3
(2𝑥𝑥 + 1)2 − (𝑥𝑥 − 3)2 = 0
𝑥𝑥1 = −4, 𝑥𝑥2 =
𝑥𝑥 = (𝑥𝑥 + 2)2
∄ 𝑥𝑥 ∈ 𝑅𝑅
𝑥𝑥 2 + �√2 − √3�𝑥𝑥 − √6 = 0
𝑥𝑥1 =
1 1
1
+ 𝑥𝑥 = −
2 6
4
𝑥𝑥1 =
𝑥𝑥 2 + 16 = 6√2𝑥𝑥
−9 − 3√5
−9 + 3√5
, 𝑥𝑥2 =
2
2
−1 − √37
−1 + √37
, 𝑥𝑥2 =
12
12
𝑥𝑥1 = 2√2, 𝑥𝑥2 = 4√2
4√5𝑥𝑥 − 1 = 4𝑥𝑥 2
𝑥𝑥1 =
𝑥𝑥 2 − 4√3𝑥𝑥 + 12 = 0
√5 − 2
√5 + 2
, 𝑥𝑥2 =
2
2
𝑥𝑥1 = 𝑥𝑥2 = 2√3
𝑥𝑥 + √5 = �2𝑥𝑥 − √5��𝑥𝑥 + √5�
𝑥𝑥1 = −√5, 𝑥𝑥2 =
2
1 2
3
�𝑥𝑥 + � = � − 2𝑥𝑥�
2
2
𝑥𝑥1 = 2, 𝑥𝑥2 =
𝑥𝑥�𝑥𝑥 + √7� = 14
v 3.0
2
3
𝑥𝑥1 = −√2, 𝑥𝑥2 = √3
𝑥𝑥2
+ 3𝑥𝑥1 + 3 = 0
3
𝑥𝑥 2 −
1 − √2
1 + √2
, 𝑥𝑥2 =
2
2
1
3
1 + √5
2
𝑥𝑥1 = −2√7, 𝑥𝑥2 = √7
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3 di 6
Equazioni di secondo grado
Algebra
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1
𝑥𝑥 2 − 1 2
(3 − 𝑥𝑥) +
− =0
3
6
3
2√3𝑥𝑥 = 𝑥𝑥 2 + 2
𝑥𝑥1 = 𝑥𝑥2 = 1
𝑥𝑥1 = √3 − 1, 𝑥𝑥2 = √3 + 1
equazioni di secondo grado frazionarie
2 𝑥𝑥
=
𝑥𝑥 2
𝑥𝑥1 = −2, 𝑥𝑥2 = 2
𝑥𝑥
√3
=
𝑥𝑥
√27
𝑥𝑥1 = −3, 𝑥𝑥2 = 3
𝑥𝑥 =
1
𝑥𝑥
𝑥𝑥 − 1 =
−
𝑥𝑥1 = −1, 𝑥𝑥2 = 1
9
𝑥𝑥 − 1
𝑥𝑥1 = −2, 𝑥𝑥2 = 4
2
1
= 2
𝑥𝑥 + 3 𝑥𝑥
∄ 𝑥𝑥 ∈ 𝑅𝑅
1
2
=
𝑥𝑥 2 𝑥𝑥 + 3
𝑥𝑥1 = −1, 𝑥𝑥2 =
2
1 − 𝑥𝑥 2
=
𝑥𝑥 2 + 1 𝑥𝑥 2 + 1
𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
3
2𝑥𝑥
=
𝑥𝑥 − 1 𝑥𝑥 + 1
𝑥𝑥1 = −
3
4𝑥𝑥 − 2
=
𝑥𝑥(𝑥𝑥 − 2)
𝑥𝑥
𝑥𝑥1 =
4
1
1 + 𝑥𝑥
𝑥𝑥
+
=
−
3 𝑥𝑥 − 1 𝑥𝑥 − 1 𝑥𝑥 + 1
1
, 𝑥𝑥 = 3
2 2
5 − √21
5 + √21
, 𝑥𝑥2 =
4
4
1
𝑥𝑥1 = − , 𝑥𝑥2 = 2
2
1
2 − 𝑥𝑥
=
2
𝑥𝑥
𝑥𝑥
𝑥𝑥1 = 𝑥𝑥2 = 1
𝑥𝑥 + 3 1
1
− 2=−
𝑥𝑥
𝑥𝑥
𝑥𝑥
𝑥𝑥1 = −2 − √5 , 𝑥𝑥2 = −2 + √5
3𝑥𝑥
𝑥𝑥 2
2𝑥𝑥
+ 2
=
𝑥𝑥 + 1 2𝑥𝑥 + 𝑥𝑥 − 1 1 − 2𝑥𝑥
𝑥𝑥1 = 0, 𝑥𝑥2 =
11 − 3𝑥𝑥
1 + 𝑥𝑥
1
= 2
−
4(𝑥𝑥 − 1) 𝑥𝑥 − 1 𝑥𝑥 + 1
1
9
1
𝑥𝑥1 = − , 𝑥𝑥2 = 3
3
1
𝑥𝑥 + 1 1 − 𝑥𝑥
+
=
𝑥𝑥 2 − 1 𝑥𝑥 − 1 𝑥𝑥 + 1
∄ 𝑥𝑥 ∈ 𝑅𝑅
2
𝑥𝑥
8
+
=
𝑥𝑥 + 1 1 − 𝑥𝑥 1 − 𝑥𝑥 2
v 3.0
3
2
𝑥𝑥1 = −2, 𝑥𝑥2 = 3
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4 di 6
Equazioni di secondo grado
Algebra
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74
75
3𝑥𝑥 + 2 1 𝑥𝑥 + 1
+ =
𝑥𝑥 2 + 2𝑥𝑥 𝑥𝑥 𝑥𝑥 + 2
𝑥𝑥1 = −1, 𝑥𝑥2 = 4
𝑥𝑥 2 − 11
𝑥𝑥
𝑥𝑥 + 1
+
=
2
𝑥𝑥 + 2𝑥𝑥 − 3 𝑥𝑥 − 1 𝑥𝑥 + 3
𝑥𝑥1 = −5, 𝑥𝑥2 = 2
2
2𝑥𝑥
2 − 𝑥𝑥
−
=
𝑥𝑥 𝑥𝑥 + 1
𝑥𝑥
𝑥𝑥1 = 1, 𝑥𝑥2 = 0 𝑛𝑛𝑛𝑛𝑛𝑛 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
equazioni di secondo grado di vario tipo
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83
84
85
86
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89
90
4𝑥𝑥(2 − 𝑥𝑥) + (𝑥𝑥 − 2) = −11
𝑥𝑥1 = −3, 𝑥𝑥2 = 3
𝑥𝑥�𝑥𝑥 + √7� = 14
𝑥𝑥1 = −2√7, 𝑥𝑥2 = √7
1
(𝑥𝑥 − 1) + 𝑥𝑥 = (𝑥𝑥 + 1)(𝑥𝑥 − 2)
2
1
𝑥𝑥1 = − , 𝑥𝑥2 = 3
2
2
�√3 − 2𝑥𝑥� = �𝑥𝑥 + √3�
2
𝑥𝑥1 = 0, 𝑥𝑥2 = 2√3
4 + (1 + 2𝑥𝑥)2 = 0
∄ 𝑥𝑥 ∈ 𝑅𝑅
(2𝑥𝑥 − 1)2 + 4(𝑥𝑥 − 2)(𝑥𝑥 + 2) = −18
∄ 𝑥𝑥 ∈ 𝑅𝑅
1
𝑥𝑥 2 − 1 2
(3 − 𝑥𝑥) +
− =0
3
6
3
2
𝑥𝑥1 = 𝑥𝑥2 = 1
√2(2𝑥𝑥 + 1) + �𝑥𝑥 − √2� = 𝑥𝑥 + 4
𝑥𝑥1 = 1 − √2, 𝑥𝑥2 = √2
1
1
1
+
=
𝑥𝑥 + 1 12 1 − 𝑥𝑥
𝑥𝑥1 = −12 − √145, 𝑥𝑥2 = −12 + √145
2√3𝑥𝑥 = 𝑥𝑥 2 + 2
1
𝑥𝑥 + √3
+
𝑥𝑥 + 2 =
1
𝑥𝑥 − √3
=3
𝑥𝑥1 =
6
+1
𝑥𝑥 + 2
1 − 2√7
1 + 2√7
, 𝑥𝑥2 =
3
3
𝑥𝑥1 = −4,
𝑥𝑥 − 5 𝑥𝑥
5
− =
8
5 𝑥𝑥 − 10
𝑥𝑥1 = −
1
4 − 𝑥𝑥
2
−
=
𝑥𝑥(2 − 𝑥𝑥) 𝑥𝑥(𝑥𝑥 + 2) 4 − 𝑥𝑥 2
10
,
3
𝑥𝑥2 = 1
𝑥𝑥2 = 5
𝑥𝑥1 = 2 𝑛𝑛𝑛𝑛𝑛𝑛 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎, 𝑥𝑥2 = 3
1
1
1
−
=
𝑥𝑥(𝑥𝑥 + 3) 𝑥𝑥(2 − 𝑥𝑥) 6𝑥𝑥
v 3.0
𝑥𝑥1 = √3 − 1, 𝑥𝑥2 = √3 + 1
𝑥𝑥1 = −1,
© 2016 - www.matematika.it
𝑥𝑥2 = 12
5 di 6
Equazioni di secondo grado
Algebra
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𝑥𝑥 + 2 (2 + 3𝑥𝑥)
4
−
= 1+ 2
𝑥𝑥
2(4 − 3𝑥𝑥)
3𝑥𝑥 − 4𝑥𝑥
𝑥𝑥1 = −6,
𝑥𝑥
1 𝑥𝑥 𝑥𝑥 + 6
− = +
𝑥𝑥 − 6 2 6 6 − 𝑥𝑥
𝑥𝑥1 = −3,
1
1
1
−
=
3𝑥𝑥 − 𝑥𝑥 2 6𝑥𝑥 2𝑥𝑥 − 𝑥𝑥 2
∄ 𝑥𝑥 ∈ 𝑅𝑅
5 9 𝑥𝑥 − 3
− =
4𝑥𝑥 4 𝑥𝑥 + 1
𝑥𝑥1 = −
4
3𝑥𝑥
+3=
2
(𝑥𝑥 − 2)
2 − 𝑥𝑥
∄ 𝑥𝑥 ∈ 𝑅𝑅
𝑥𝑥 − √3
5
,
13
𝑥𝑥2 =
4
𝑛𝑛𝑛𝑛𝑛𝑛 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
3
𝑥𝑥2 = 1
𝑥𝑥2 = 18
2√3
𝑥𝑥
−
2
3𝑥𝑥 − 1 𝑥𝑥√3 + 1
𝑥𝑥1 = −
5
1
3𝑥𝑥 + 2
+
= 2
2𝑥𝑥 + 2 2𝑥𝑥 − 2 𝑥𝑥 + 𝑥𝑥 + 1
𝑥𝑥1 = 0,
2
1
1
1 2
− �𝑥𝑥 −
� �𝑥𝑥 −
� + �𝑥𝑥 −
� =0
(𝑥𝑥 − 1)2
𝑥𝑥 − 1
𝑥𝑥 + 1
𝑥𝑥 − 1
𝑥𝑥1 = 1 − √3, 𝑥𝑥2 = 1 + √3
𝑥𝑥√3 − 1
=
4
3
5
− 2=
2
(𝑥𝑥 − 2)
𝑥𝑥
4𝑥𝑥 − 2𝑥𝑥 2
𝑥𝑥1 =
√3
, 𝑥𝑥 = √3
2 2
−7 − √217
,
7
𝑥𝑥2 =
𝑥𝑥2 = 4
5
1
5
− 2
=
6 + 2𝑥𝑥 𝑥𝑥 − 3𝑥𝑥 9 − 𝑥𝑥 2
3
𝑥𝑥1 = − ,
5
1
1
2
1
−
=
+
𝑥𝑥 − 5 4 − 𝑥𝑥 𝑥𝑥 − 4 𝑥𝑥 − 3
𝑥𝑥1 = 5 − √2, 𝑥𝑥2 = 5 + √2
𝑥𝑥 2
1
=
2
200 − 𝑥𝑥
7
𝑥𝑥1 = −5,
𝑥𝑥 + 3 𝑥𝑥 + 5 2𝑥𝑥 + 2
+
=
𝑥𝑥 + 1 𝑥𝑥 + 2
𝑥𝑥
4
𝑥𝑥1 = − ,
3
2(𝑥𝑥 − 1) (𝑥𝑥 + 1)
+
=3
𝑥𝑥 + 1
4(𝑥𝑥 − 1)
𝑥𝑥1 =
𝑥𝑥 − 2 2𝑥𝑥 + 9 2𝑥𝑥 + 1
+
=
3 − 𝑥𝑥 𝑥𝑥 2 − 9
𝑥𝑥 + 3
𝑥𝑥2 = 2
𝑥𝑥2 = 1
𝑥𝑥2 = 5
−7 − 4√7
−7 + 4√7
, 𝑥𝑥2 =
3
3
𝑥𝑥1 = 1 − √7, 𝑥𝑥2 = 1 + √7
5𝑥𝑥 2 − 5𝑥𝑥 − 28
4
2
=
−
𝑥𝑥 2 − 𝑥𝑥 − 6
𝑥𝑥 + 2 𝑥𝑥 − 3
v 3.0
−7 + √217
7
𝑥𝑥1 = −1,
© 2016 - www.matematika.it
𝑥𝑥2 =
12
5
6 di 6