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Byju's Answer
Standard XII
Mathematics
Roots of a Quadratic Equation
Find the quad...
Question
Find the quadratic equation with rational coefficients which has
5
−
2
√
5
as a root.
A
x
2
−
10
x
−
5
=
0
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B
x
2
+
10
x
+
5
=
0
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C
x
2
−
10
x
+
5
=
0
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D
None of the above
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Solution
The correct option is
C
x
2
−
10
x
+
5
=
0
According to the problem, coefficients of the required quadratic equation are rational and its one root is
5
−
2
√
5
.
We know in a quadratic equation with rational coefficients irrational roots occur in conjugate pairs.
Since equation has rational coefficients, the other root is
5
+
2
√
5
.
Now, sum of roots
=
S
=
(
5
−
2
√
5
)
+
(
5
+
2
√
5
)
=
10
and product of roots
=
P
=
(
5
−
2
√
5
)
(
5
+
2
√
5
)
=
25
−
(
2
√
5
)
2
=
25
−
20
=
5
Hence, required quadratic equation is
x
2
−
S
x
+
P
=
0
i.e.
x
2
−
10
x
+
5
=
0
.
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0
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