מתוך ויקיספר, אוסף הספרים והמדריכים החופשי
[ עריכה ] פתרו את התרגילים הבאים בעזרת נוסחת השורשים של משוואה ריבועית. מותר ועדיף להשתמש במחשבון. מומלץ לצמצם גורמים משותפים לפני הצבה בנוסחת השורשים על-ידי חילוק בהם.
7
x
2
−
115
12
x
−
389
14
=
6
x
2
−
7
3
x
−
2
7
{\displaystyle 7x^{2}-{\frac {115}{12}}x-{\frac {389}{14}}=6x^{2}-{\frac {7}{3}}x-{\frac {2}{7}}}
3
2
x
2
+
29
10
x
−
42
5
=
1
2
x
2
−
8
5
x
+
3
5
{\displaystyle {\frac {3}{2}}x^{2}+{\frac {29}{10}}x-{\frac {42}{5}}={\frac {1}{2}}x^{2}-{\frac {8}{5}}x+{\frac {3}{5}}}
x
2
+
97
12
x
+
211
24
=
4
3
x
−
4
3
{\displaystyle x^{2}+{\frac {97}{12}}x+{\frac {211}{24}}={\frac {4}{3}}x-{\frac {4}{3}}}
2
x
2
+
4
3
x
−
19
12
=
7
4
{\displaystyle 2x^{2}+{\frac {4}{3}}x-{\frac {19}{12}}={\frac {7}{4}}}
4
x
2
−
22
x
+
158
5
=
8
5
{\displaystyle 4x^{2}-22x+{\frac {158}{5}}={\frac {8}{5}}}
2
x
2
+
14
x
+
85
4
=
−
5
4
{\displaystyle 2x^{2}+14x+{\frac {85}{4}}=-{\frac {5}{4}}}
2
3
x
2
−
5
4
x
−
31
2
=
−
4
3
x
2
−
5
4
x
−
3
{\displaystyle {\frac {2}{3}}x^{2}-{\frac {5}{4}}x-{\frac {31}{2}}=-{\frac {4}{3}}x^{2}-{\frac {5}{4}}x-3}
2
x
2
−
67
4
x
−
107
2
=
3
4
x
−
4
{\displaystyle 2x^{2}-{\frac {67}{4}}x-{\frac {107}{2}}={\frac {3}{4}}x-4}
35
8
x
2
+
19
x
+
28
=
3
8
x
2
−
2
x
+
1
2
{\displaystyle {\frac {35}{8}}x^{2}+19x+28={\frac {3}{8}}x^{2}-2x+{\frac {1}{2}}}
2
x
2
+
9
4
x
−
113
30
=
x
2
+
1
6
x
+
2
5
{\displaystyle 2x^{2}+{\frac {9}{4}}x-{\frac {113}{30}}=x^{2}+{\frac {1}{6}}x+{\frac {2}{5}}}
x
2
+
7
6
x
−
22
3
=
1
2
x
−
1
3
{\displaystyle x^{2}+{\frac {7}{6}}x-{\frac {22}{3}}={\frac {1}{2}}x-{\frac {1}{3}}}
4
x
2
+
14
x
−
22
3
=
2
3
{\displaystyle 4x^{2}+14x-{\frac {22}{3}}={\frac {2}{3}}}
2
x
2
+
50
3
x
−
355
4
=
−
3
4
{\displaystyle 2x^{2}+{\frac {50}{3}}x-{\frac {355}{4}}=-{\frac {3}{4}}}
1
2
x
2
+
8
x
+
11
3
=
−
7
2
x
2
−
4
3
x
+
1
{\displaystyle {\frac {1}{2}}x^{2}+8x+{\frac {11}{3}}=-{\frac {7}{2}}x^{2}-{\frac {4}{3}}x+1}
3
x
2
+
149
10
x
−
239
14
=
7
5
x
−
4
7
{\displaystyle 3x^{2}+{\frac {149}{10}}x-{\frac {239}{14}}={\frac {7}{5}}x-{\frac {4}{7}}}
31
8
x
2
−
162
7
x
−
60
=
7
8
x
2
+
6
7
x
{\displaystyle {\frac {31}{8}}x^{2}-{\frac {162}{7}}x-60={\frac {7}{8}}x^{2}+{\frac {6}{7}}x}
4
x
2
+
10
x
−
104
=
−
2
x
+
8
{\displaystyle 4x^{2}+10x-104=-2x+8}
8
7
x
2
+
46
5
x
+
33
4
=
−
6
7
x
2
+
6
5
x
+
1
4
{\displaystyle {\frac {8}{7}}x^{2}+{\frac {46}{5}}x+{\frac {33}{4}}=-{\frac {6}{7}}x^{2}+{\frac {6}{5}}x+{\frac {1}{4}}}
−
2
5
x
2
+
1
2
x
−
15
=
−
7
5
x
2
+
1
2
x
+
1
{\displaystyle -{\frac {2}{5}}x^{2}+{\frac {1}{2}}x-15=-{\frac {7}{5}}x^{2}+{\frac {1}{2}}x+1}
x
2
+
11
4
x
+
45
28
=
−
1
7
{\displaystyle x^{2}+{\frac {11}{4}}x+{\frac {45}{28}}=-{\frac {1}{7}}}
x
2
−
7
3
x
−
2
3
=
−
2
{\displaystyle x^{2}-{\frac {7}{3}}x-{\frac {2}{3}}=-2}
4
x
2
+
12
x
−
214
3
=
2
3
{\displaystyle 4x^{2}+12x-{\frac {214}{3}}={\frac {2}{3}}}
4
x
2
+
8
x
+
2
=
−
2
{\displaystyle 4x^{2}+8x+2=-2}
x
2
−
13
2
x
+
53
5
=
3
5
{\displaystyle x^{2}-{\frac {13}{2}}x+{\frac {53}{5}}={\frac {3}{5}}}
x
2
+
91
15
x
+
57
8
=
2
5
x
−
7
8
{\displaystyle x^{2}+{\frac {91}{15}}x+{\frac {57}{8}}={\frac {2}{5}}x-{\frac {7}{8}}}
3
2
x
2
+
8
3
x
−
161
8
=
−
5
2
x
2
−
4
3
x
−
7
8
{\displaystyle {\frac {3}{2}}x^{2}+{\frac {8}{3}}x-{\frac {161}{8}}=-{\frac {5}{2}}x^{2}-{\frac {4}{3}}x-{\frac {7}{8}}}
13
4
x
2
−
4
x
−
5
=
−
3
4
x
2
+
x
+
4
{\displaystyle {\frac {13}{4}}x^{2}-4x-5=-{\frac {3}{4}}x^{2}+x+4}
x
2
+
21
4
x
−
69
4
=
−
5
{\displaystyle x^{2}+{\frac {21}{4}}x-{\frac {69}{4}}=-5}
x
2
−
73
14
x
−
153
8
=
2
7
x
+
7
8
{\displaystyle x^{2}-{\frac {73}{14}}x-{\frac {153}{8}}={\frac {2}{7}}x+{\frac {7}{8}}}
9
x
2
−
47
6
x
−
28
=
5
x
2
+
7
6
x
{\displaystyle 9x^{2}-{\frac {47}{6}}x-28=5x^{2}+{\frac {7}{6}}x}
2
x
2
+
38
3
x
+
389
21
=
−
1
7
{\displaystyle 2x^{2}+{\frac {38}{3}}x+{\frac {389}{21}}=-{\frac {1}{7}}}
4
x
2
+
32
3
x
−
7
2
=
−
7
2
{\displaystyle 4x^{2}+{\frac {32}{3}}x-{\frac {7}{2}}=-{\frac {7}{2}}}
3
x
2
−
6
x
+
7
4
=
−
1
2
{\displaystyle 3x^{2}-6x+{\frac {7}{4}}=-{\frac {1}{2}}}
7
3
x
2
−
26
7
x
+
5
8
=
−
2
3
x
2
−
5
7
x
+
5
8
{\displaystyle {\frac {7}{3}}x^{2}-{\frac {26}{7}}x+{\frac {5}{8}}=-{\frac {2}{3}}x^{2}-{\frac {5}{7}}x+{\frac {5}{8}}}
x
2
−
32
3
x
+
172
21
=
−
5
x
+
6
7
{\displaystyle x^{2}-{\frac {32}{3}}x+{\frac {172}{21}}=-5x+{\frac {6}{7}}}
x
2
−
4
3
x
−
4
=
1
{\displaystyle x^{2}-{\frac {4}{3}}x-4=1}
2
x
2
−
6
x
−
35
3
=
4
3
x
+
5
3
{\displaystyle 2x^{2}-6x-{\frac {35}{3}}={\frac {4}{3}}x+{\frac {5}{3}}}
30
7
x
2
+
33
2
x
−
195
4
=
2
7
x
2
+
1
2
x
−
3
4
{\displaystyle {\frac {30}{7}}x^{2}+{\frac {33}{2}}x-{\frac {195}{4}}={\frac {2}{7}}x^{2}+{\frac {1}{2}}x-{\frac {3}{4}}}
2
x
2
+
86
3
x
+
73
=
−
4
3
x
+
1
{\displaystyle 2x^{2}+{\frac {86}{3}}x+73=-{\frac {4}{3}}x+1}
3
x
2
−
21
x
+
37
=
1
{\displaystyle 3x^{2}-21x+37=1}
4
x
2
−
46
x
+
786
7
=
2
7
{\displaystyle 4x^{2}-46x+{\frac {786}{7}}={\frac {2}{7}}}
x
2
+
67
5
x
+
31
=
−
3
5
x
−
2
{\displaystyle x^{2}+{\frac {67}{5}}x+31=-{\frac {3}{5}}x-2}
4
x
2
−
8
21
x
−
140
3
=
2
7
x
{\displaystyle 4x^{2}-{\frac {8}{21}}x-{\frac {140}{3}}={\frac {2}{7}}x}
3
x
2
−
281
28
x
+
25
8
=
−
2
7
x
−
5
2
{\displaystyle 3x^{2}-{\frac {281}{28}}x+{\frac {25}{8}}=-{\frac {2}{7}}x-{\frac {5}{2}}}
4
x
2
+
42
x
+
26
=
6
{\displaystyle 4x^{2}+42x+26=6}
x
2
+
7
3
x
−
79
21
=
−
3
7
{\displaystyle x^{2}+{\frac {7}{3}}x-{\frac {79}{21}}=-{\frac {3}{7}}}
17
4
x
2
−
10
3
x
−
1081
21
=
1
4
x
2
−
4
x
−
1
7
{\displaystyle {\frac {17}{4}}x^{2}-{\frac {10}{3}}x-{\frac {1081}{21}}={\frac {1}{4}}x^{2}-4x-{\frac {1}{7}}}
x
2
−
5
4
x
−
27
2
=
−
1
2
x
−
8
{\displaystyle x^{2}-{\frac {5}{4}}x-{\frac {27}{2}}=-{\frac {1}{2}}x-8}
−
40
3
x
+
25
3
=
−
x
2
−
5
3
x
+
1
{\displaystyle -{\frac {40}{3}}x+{\frac {25}{3}}=-x^{2}-{\frac {5}{3}}x+1}
4
x
2
+
43
3
x
+
13
3
=
4
3
x
+
4
3
{\displaystyle 4x^{2}+{\frac {43}{3}}x+{\frac {13}{3}}={\frac {4}{3}}x+{\frac {4}{3}}}
{
−
11
4
,
10
}
{\displaystyle \left\{-{\frac {11}{4}},10\right\}}
{
−
6
,
3
2
}
{\displaystyle \left\{-6,{\frac {3}{2}}\right\}}
{
−
9
4
,
−
9
2
}
{\displaystyle \left\{-{\frac {9}{4}},-{\frac {9}{2}}\right\}}
{
−
5
3
,
1
}
{\displaystyle \left\{-{\frac {5}{3}},1\right\}}
{
5
2
,
3
}
{\displaystyle \left\{{\frac {5}{2}},3\right\}}
{
−
9
2
,
−
5
2
}
{\displaystyle \left\{-{\frac {9}{2}},-{\frac {5}{2}}\right\}}
{
−
5
2
,
5
2
}
{\displaystyle \left\{-{\frac {5}{2}},{\frac {5}{2}}\right\}}
{
−
9
4
,
11
}
{\displaystyle \left\{-{\frac {9}{4}},11\right\}}
{
−
11
4
,
−
5
2
}
{\displaystyle \left\{-{\frac {11}{4}},-{\frac {5}{2}}\right\}}
{
5
4
,
−
10
3
}
{\displaystyle \left\{{\frac {5}{4}},-{\frac {10}{3}}\right\}}
{
−
3
,
7
3
}
{\displaystyle \left\{-3,{\frac {7}{3}}\right\}}
{
−
4
,
1
2
}
{\displaystyle \left\{-4,{\frac {1}{2}}\right\}}
{
−
12
,
11
3
}
{\displaystyle \left\{-12,{\frac {11}{3}}\right\}}
{
−
2
,
−
1
3
}
{\displaystyle \left\{-2,-{\frac {1}{3}}\right\}}
{
1
,
−
11
2
}
{\displaystyle \left\{1,-{\frac {11}{2}}\right\}}
{
10
,
−
2
}
{\displaystyle \{10,-2\}}
{
−
7
,
4
}
{\displaystyle \{-7,4\}}
{
−
2
}
{\displaystyle \{-2\}}
{
4
,
−
4
}
{\displaystyle \{4,-4\}}
{
−
7
4
,
−
1
}
{\displaystyle \left\{-{\frac {7}{4}},-1\right\}}
{
1
,
4
3
}
{\displaystyle \left\{1,{\frac {4}{3}}\right\}}
{
−
6
,
3
}
{\displaystyle \{-6,3\}}
{
−
1
}
{\displaystyle \{-1\}}
{
5
2
,
4
}
{\displaystyle \left\{{\frac {5}{2}},4\right\}}
{
−
3
,
−
8
3
}
{\displaystyle \left\{-3,-{\frac {8}{3}}\right\}}
{
7
4
,
−
11
4
}
{\displaystyle \left\{{\frac {7}{4}},-{\frac {11}{4}}\right\}}
{
−
1
,
9
4
}
{\displaystyle \left\{-1,{\frac {9}{4}}\right\}}
{
−
7
,
7
4
}
{\displaystyle \left\{-7,{\frac {7}{4}}\right\}}
{
−
5
2
,
8
}
{\displaystyle \left\{-{\frac {5}{2}},8\right\}}
{
−
7
4
,
4
}
{\displaystyle \left\{-{\frac {7}{4}},4\right\}}
{
−
7
3
,
−
4
}
{\displaystyle \left\{-{\frac {7}{3}},-4\right\}}
{
−
8
3
,
0
}
{\displaystyle \left\{-{\frac {8}{3}},0\right\}}
{
3
2
,
1
2
}
{\displaystyle \left\{{\frac {3}{2}},{\frac {1}{2}}\right\}}
{
0
,
1
}
{\displaystyle \{0,1\}}
{
2
,
11
3
}
{\displaystyle \left\{2,{\frac {11}{3}}\right\}}
{
3
,
−
5
3
}
{\displaystyle \left\{3,-{\frac {5}{3}}\right\}}
{
−
4
3
,
5
}
{\displaystyle \left\{-{\frac {4}{3}},5\right\}}
{
2
,
−
6
}
{\displaystyle \{2,-6\}}
{
−
12
,
−
3
}
{\displaystyle \{-12,-3\}}
{
3
,
4
}
{\displaystyle \{3,4\}}
{
7
2
,
8
}
{\displaystyle \left\{{\frac {7}{2}},8\right\}}
{
−
3
,
−
11
}
{\displaystyle \{-3,-11\}}
{
7
2
,
−
10
3
}
{\displaystyle \left\{{\frac {7}{2}},-{\frac {10}{3}}\right\}}
{
5
2
,
3
4
}
{\displaystyle \left\{{\frac {5}{2}},{\frac {3}{4}}\right\}}
{
−
1
2
,
−
10
}
{\displaystyle \left\{-{\frac {1}{2}},-10\right\}}
{
1
,
−
10
3
}
{\displaystyle \left\{1,-{\frac {10}{3}}\right\}}
{
7
2
,
−
11
3
}
{\displaystyle \left\{{\frac {7}{2}},-{\frac {11}{3}}\right\}}
{
−
2
,
11
4
}
{\displaystyle \left\{-2,{\frac {11}{4}}\right\}}
{
2
3
,
11
}
{\displaystyle \left\{{\frac {2}{3}},11\right\}}
{
−
3
,
−
1
4
}
{\displaystyle \left\{-3,-{\frac {1}{4}}\right\}}
