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Joint Entrance Examination
Mathematics
Complex Numbers
Question 255
Question 255 of 455
Medium
MCQ
If
α
\alpha
α
and
β
\beta
β
are the roots of
2
x
(
2
x
+
1
)
=
1
2x(2x+1)=1
2
x
(
2
x
+
1
)
=
1
, then
β
\beta
β
is equal to:
A
2
α
(
α
+
1
)
2\alpha(\alpha+1)
2
α
(
α
+
1
)
B
−
2
α
(
α
+
1
)
-2\alpha(\alpha+1)
−
2
α
(
α
+
1
)
C
2
α
(
α
−
1
)
2\alpha(\alpha-1)
2
α
(
α
−
1
)
D
2
α
2
2\alpha^2
2
α
2
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Official solution
The equation is
4
x
2
+
2
x
−
1
=
0
4x^2+2x-1=0
4
x
2
+
2
x
−
1
=
0
, so
α
+
β
=
−
1
2
\alpha+\beta=-\dfrac12
α
+
β
=
−
2
1
, and
4
α
2
+
2
α
−
1
=
0
4\alpha^2+2\alpha-1=0
4
α
2
+
2
α
−
1
=
0
. Combining:
4
α
2
+
2
α
+
2
α
+
2
β
=
0
⟹
β
=
−
2
α
(
α
+
1
)
.
4\alpha^2+2\alpha+2\alpha+2\beta=0 \implies \beta=-2\alpha(\alpha+1).
4
α
2
+
2
α
+
2
α
+
2
β
=
0
⟹
β
=
−
2
α
(
α
+
1
)
.
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