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Byju's Answer
Standard XII
Mathematics
Relation between Roots and Coefficients for Quadratic
if alpha;...
Question
if~~α and ~~β are the roots of the equation x^2 -p (x+1) -q = 0 then value of
(α^2+2α+1)/α^2+2α+q) +(β^2+2β+1)/(β^2+2 β+q) is ?
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Solution
Dear student
Given
:
α
and
β
are
the
roots
of
x
2
-
p
x
+
1
-
q
=
0
i
.
e
.
x
2
-
px
-
p
-
q
=
0
=
x
2
-
px
-
p
+
q
=
0
α
+
β
=
-
coeff
.
of
x
coeff
.
of
x
2
=
-
-
p
1
=
p
.
.
.
1
αβ
=
constant
term
coeff
.
of
x
2
=
-
p
+
q
1
=
-
p
+
q
⇒
αβ
=
-
p
-
q
⇒
q
=
-
p
-
αβ
⇒
q
=
-
α
+
β
-
αβ
.
.
.
2
using
1
To
find
the
value
of
:
α
2
+
2
α
+
1
α
2
+
2
α
+
q
+
β
2
+
2
β
+
1
β
2
+
2
β
+
q
=
α
+
1
2
α
2
+
2
α
-
α
-
β
-
αβ
+
β
+
1
2
β
2
+
2
β
-
α
-
β
-
αβ
using
2
=
α
+
1
2
α
2
+
α
-
β
-
αβ
+
β
+
1
2
β
2
+
β
-
α
-
αβ
=
α
+
1
2
α
-
β
α
+
1
+
β
+
1
2
β
-
α
β
+
1
=
α
+
1
α
-
β
-
β
+
1
α
-
β
=
α
+
1
-
β
-
1
α
-
β
=
α
-
β
α
-
β
=
1
Note
:
a
±
b
2
=
a
2
+
b
2
±
2
ab
Regards
Suggest Corrections
0
Similar questions
Q.
Let
α
,
β
are roots of the equation
x
2
−
p
(
x
+
1
)
−
q
=
0
,
then the value of
α
2
+
2
α
+
1
α
2
+
2
α
+
q
+
β
2
+
2
β
+
1
β
2
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β
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q
is
Q.
If
α
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β
are the roots of
x
2
−
p
(
x
+
1
)
−
c
=
0
, then the value of
α
2
+
2
α
+
1
α
2
+
2
α
+
c
+
β
2
+
2
β
+
1
β
2
+
2
β
+
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is
Q.
If
α
,
β
are the roots of the equation
x
2
−
p
(
x
+
1
)
−
c
=
0
, then the value of
α
2
+
2
α
+
1
α
2
+
2
α
+
c
+
β
2
+
2
β
+
1
β
2
+
2
β
+
c
is
Q.
lf
α
,
β
are the roots of
x
2
−
p
x
−
c
−
p
=
0
, then
α
2
+
2
α
+
1
α
2
+
2
α
+
c
+
β
2
+
2
β
+
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β
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+
2
β
+
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=
Q.
If
α
,
β
are roots of
x
2
−
p
(
x
+
1
)
−
c
=
0
. Show that
i)
(
α
+
1
)
(
β
+
1
)
=
1
−
c
&
ii)
α
2
+
2
α
+
1
α
2
+
2
α
+
c
+
β
2
+
2
β
+
1
β
2
+
2
β
+
c
=
1
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