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Byju's Answer
Standard XII
Mathematics
Geometric Mean
If α ,β ,γ ...
Question
If
α
,
β
,
γ
are the roots of
x
3
−
6
x
−
4
=
0
, then the equation whose roots are
β
γ
+
1
α
,
γ
α
+
1
β
,
α
β
+
1
γ
is
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Solution
Given equation:
x
3
−
6
x
−
4
=
0
roots of the given equation is given by
α
+
β
+
γ
=
0
α
β
+
β
γ
+
γ
α
=
−
6
α
β
γ
=
4
now the value of roots given in the question are
α
β
+
1
γ
=
α
β
γ
+
1
γ
=
5
γ
α
γ
+
1
β
=
α
β
γ
+
1
β
=
5
β
β
γ
+
1
α
=
α
β
γ
+
1
α
=
5
α
Now calculating the sum of the roots, we get
5
γ
+
5
β
+
5
α
=
−
15
2
Now calculating product of the roots, we get
5
γ
∗
5
β
∗
5
α
=
125
4
Now calculating sum of the products of the roots, we get
5
γ
∗
5
β
+
5
β
∗
5
α
+
5
γ
∗
5
α
=
0
Therefore the general cubic equation is
x
3
+(Sum of the roots)
x
2
+
(sum of the products of the roots)
x
+
(products of the roots)
=
0
x
3
+
(
−
15
2
)
x
2
+
(
0
)
x
+
(
125
4
)
=
0
4
x
3
−
30
x
2
+
125
=
0
which is the required equation.
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0
Similar questions
Q.
If
α
,
β
,
γ
are non zero roots of
x
3
+
p
x
2
+
q
x
+
r
=
0
, then the equation whose roots are
α
(
β
+
γ
)
,
β
(
γ
+
α
)
,
γ
(
α
+
β
)
Q.
If
α
,
β
,
γ
are the roots of the equation
x
3
+
p
x
2
+
q
x
+
r
=
0
then the coefficient of
x
in the cubic equation whose roots are
α
(
β
+
γ
)
,
β
(
γ
+
α
)
and
γ
(
α
+
β
)
is
Q.
If
α
,
β
,
γ
are the roots of the equation
2
x
3
−
5
x
2
+
3
x
−
1
=
0
,
then
1
α
β
+
1
β
γ
+
1
γ
α
=
Q.
If
2
x
3
+
3
x
2
+
5
x
+
6
=
0
has roots
α
,
β
,
γ
then find
α
+
β
+
γ
,
α
β
+
β
γ
+
γ
α
and
α
β
γ
Q.
If
α
,
β
and
γ
are the roots of the cubic equation
3
x
3
+
2
x
2
+
x
−
1
=
0
, then the value of
(
α
β
+
β
γ
+
γ
α
)
is:
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