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Byju's Answer
Standard XII
Mathematics
Common Roots
#945; #160;...
Question
α
a
n
d
β
a
r
e
t
h
e
r
o
o
t
s
o
f
t
h
e
e
q
u
.
a
x
2
+
b
x
+
c
=
0
a
n
d
α
4
,
β
4
a
r
e
t
h
e
r
o
o
t
s
o
f
t
h
e
e
q
u
a
t
i
o
n
l
x
2
+
m
x
+
n
=
0
(
α
,
β
a
r
e
r
e
a
l
a
n
d
d
i
s
t
i
n
c
t
)
.
L
e
t
f
(
x
)
=
a
2
l
x
2
-
4
a
c
l
x
-
2
c
2
l
+
a
2
m
=
0
t
h
e
n
O
n
e
r
o
o
t
o
f
t
h
e
f
(
x
)
=
0
i
s
i
.
8
2
b
2
i
i
.
b
2
a
2
i
i
i
.
b
/
a
i
v
.
a
/
b
Open in App
Solution
Dear student
NOTE
:
If
px
2
+
qx
+
r
is
a
quadratic
equations
.
Then
Sum
of
roots
=
-
Coeff
.
of
x
Coeff
.
of
x
2
=
-
q
p
and
product
of
roots
=
Constant
term
Coeff
.
of
x
2
=
r
p
Since
α
and
β
are
the
roots
of
ax
2
+
bx
+
c
=
0
.
Then
α
+
β
=
-
b
a
and
αβ
=
c
a
.
.
.
.
(
1
)
Also
,
α
4
and
β
4
are
the
roots
of
lx
2
+
mx
+
n
=
0
Then
α
4
+
β
4
=
-
m
l
.
.
.
(
2
)
and
α
4
β
4
=
n
l
Now
,
Consider
,
f
(
x
)
=
a
2
lx
2
-
4
aclx
+
2
c
2
l
+
a
2
m
=
0
Then
x
=
4
acl
±
4
acl
2
-
4
a
2
l
2
c
2
l
+
a
2
m
2
a
2
l
⇒
x
=
4
acl
±
16
a
2
c
2
l
2
-
8
a
2
c
2
l
2
-
4
a
4
lm
2
a
2
l
⇒
x
=
4
acl
±
8
a
2
c
2
l
2
-
4
a
4
lm
2
a
2
l
⇒
x
=
4
acl
±
2
al
2
c
2
-
a
2
m
l
2
a
2
l
⇒
x
=
2
c
a
±
1
a
2
c
2
a
2
a
2
-
a
2
m
l
⇒
x
=
2
c
a
±
a
a
2
c
2
a
2
-
m
l
⇒
x
=
2
αβ
±
2
α
2
β
2
+
α
4
+
β
4
using
1
and
2
⇒
x
=
2
αβ
±
α
2
+
β
2
2
⇒
x
=
2
αβ
±
α
2
+
β
2
⇒
x
=
2
αβ
±
α
+
β
2
-
2
αβ
∵
x
+
y
2
=
x
2
+
y
2
+
2
xy
⇒
x
=
2
αβ
+
α
+
β
2
-
2
αβ
or
x
=
2
αβ
-
α
+
β
2
-
2
αβ
⇒
x
=
α
+
β
2
or
x
=
4
αβ
-
α
+
β
2
⇒
x
=
b
2
a
2
or
x
=
4
c
a
-
b
2
a
2
⇒
x
=
b
2
a
2
or
x
=
4
ac
-
b
2
a
2
So
,
one
of
the
root
of
f
(
x
)
=
0
is
b
2
a
2
Regards
Suggest Corrections
0
Similar questions
Q.
α
and
β
are the roots of the equation
a
x
2
+
b
x
+
c
=
0
and
α
4
,
β
4
are the roots of the equation
l
x
2
+
m
x
+
n
=
0
(
α
,
β
are real and distinct.) Let
f
(
x
)
=
a
2
l
x
2
−
4
a
c
l
x
+
2
c
2
l
+
a
2
m
=
0
, then
Roots of
f
(
x
)
=
0
are
Q.
If
α
,
β
are real roots of the equation
a
x
2
+
b
x
+
c
=
0
and
α
4
,
β
4
are roots of
l
x
2
+
m
x
+
n
=
0
, then the roots of the equation
a
2
l
x
2
−
4
a
c
l
x
+
2
c
2
l
+
a
2
m
=
0
are
Q.
If
f
(
x
)
=
a
x
2
+
b
x
+
c
,
a
,
b
,
c
∈
R
and equation
f
(
x
)
−
x
=
0
has non-real roots
α
,
β
. Let
γ
,
δ
be the roots of
f
(
f
(
x
)
)
−
x
=
0
(
γ
,
δ
are not equal to
α
,
β
). Then
∣
∣ ∣
∣
2
α
δ
β
0
α
γ
β
1
∣
∣ ∣
∣
is
Q.
If roots of the equation
f
(
x
+
2
)
=
a
x
2
+
b
x
+
c
=
0
and
α
,
β
are such that
α
<
−
2
<
β
, then for the equation
f
(
x
)
=
A
x
2
+
B
x
+
C
.
Q.
Let
α
,
β
be the roots of the equation
a
x
2
+
b
x
+
c
=
0
and
α
4
+
β
4
be the roots of the equation
p
x
2
+
q
x
+
r
=
0
, then the roots of the equation
a
2
p
x
2
−
4
a
c
p
x
+
2
c
2
p
+
a
2
q
=
0
are:
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