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Byju's Answer
Standard XII
Mathematics
Nature of Roots
if alpha and ...
Question
if alpha and beta are the roots of 2x
2
+ x+3 =0, then equation whose roots are 1-alpha/1+alpha and 1-beta/1+beta is
Open in App
Solution
Dear
Student
,
2
x
2
+
x
+
3
=
0
Here
,
roots
are
α
and
β
,
Now
we
know
that
;
α
+
β
=
-
b
a
=
-
1
2
.
.
.
.
.
.
.
.
.
(
i
)
and
,
αβ
=
c
a
=
3
2
.
.
.
.
.
.
.
(
ii
)
Now
,
The
new
quadratic
equation
has
roots
which
are
;
1
-
α
1
+
α
and
1
-
β
1
+
β
Now
,
Summation
of
roots
;
1
-
α
1
+
α
+
1
-
β
1
+
β
=
(
1
+
β
)
(
1
-
α
)
+
(
1
-
β
)
(
1
+
α
)
(
1
+
α
)
(
1
+
β
)
=
1
+
β
-
α
-
αβ
+
1
+
α
-
β
-
αβ
1
+
α
+
β
+
αβ
=
2
(
1
-
αβ
)
1
+
(
α
+
β
)
+
αβ
Now
put
value
of
(
α
+
β
)
and
αβ
from
(
i
)
and
(
ii
)
,
we
get
;
=
2
(
1
-
3
2
)
1
-
1
2
+
3
2
=
-
1
2
Summation
of
roots
of
new
quadratic
equation
=
-
1
2
Now
,
(
1
-
α
1
+
α
)
×
(
1
-
β
1
+
β
)
=
1
-
α
-
β
+
αβ
1
+
α
+
β
+
αβ
=
1
-
(
α
+
β
)
+
αβ
1
+
(
α
+
β
)
+
αβ
Now
put
value
of
(
α
+
β
)
and
αβ
from
(
i
)
and
(
ii
)
,
we
get
;
=
1
+
1
2
+
3
2
1
-
1
2
+
3
2
=
3
2
Product
of
roots
of
new
quadratic
equation
=
3
2
We
can
write
quadratic
equation
in
the
below
mentioned
form
;
x
2
-
(
sum
of
roots
)
x
+
(
product
of
roots
)
=
0
So
,
⇒
x
2
-
(
-
1
2
)
x
+
3
2
=
0
⇒
2
x
2
+
x
+
3
=
0
Which
is
identical
as
parent
quadratic
equation
.
Regards
.
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0
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