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Byju's Answer
Standard X
Mathematics
Nature of Roots
Show that the...
Question
Show that the equation
x
2
+
a
x
-
1
=
0
has real and distinct roots for all real values of a.
Open in App
Solution
Given:
x
2
+
a
x
−
1
=
0
Here,
a
=
1
,
b
=
a
and
c
=
−
1
Discriminant
D
is
given
by
:
D
=
(
b
2
−
4
a
c
)
⇒
D
=
a
2
−
4
×
1
×
(
−
1
)
⇒
D
=
(
a
2
+
4
)
F
o
r
a
l
l
r
e
a
l
v
a
l
u
e
s
o
f
a
,
w
e
h
a
v
e
a
2
>
0
o
r
,
a
2
+
4
>
0
∴
D
>
0
for all real values of
a
.
Thus, the roots of the equation are real and distinct for all real values of
a
.
Suggest Corrections
1
Similar questions
Q.
For the equation
4
x
2
+
x
+
4
x
2
+
1
+
x
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x
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+
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+
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=
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6
Which of the following statement(s) is/are correct?
Q.
Show that the roots of the equation
x
2
+
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x
−
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=
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Q.
If
x
2
+
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a
−
b
)
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+
(
1
−
a
−
b
)
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,
where
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∈
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the value of
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such that the equation has distinct real roots for all value of
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are
Q.
If the equation
(
x
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+
x
2
)
2
+
a
(
x
1
+
x
2
)
+
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=
0
has exactly two real roots which are distinct , then the set of all possible real values of
a
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∪
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,
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)
then
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+
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is equal to:
Q.
Assertion :The equation
f
(
x
)
(
f
′′
(
x
)
)
2
+
f
(
x
)
f
′
(
x
)
f
′′′
(
x
)
+
(
f
′
(
x
)
)
2
f
′′
(
x
)
=
0
has atleast 5 real roots Reason: The equation
f
(
x
)
=
0
has atleast 3 real distinct roots & if
f
(
x
)
=
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has k real distinct roots, then
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′
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x
)
=
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has atleast k-1 distinct roots.
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