The factors of x²– 2V3x +3 are
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Zeros of
a
x
2
+
b
x
+
c
are given by quadratic formula
−
b
±
√
b
2
−
4
a
c
2
a
, however, such a quadratic function can be factorized, if the discriminant
(
b
2
−
4
a
c
)
is square of a rational number.
In
x
2
+
2
x
+
3
, discriminant is
2
2
−
4
⋅
1
⋅
3
=
4
−
12
=
−
8
and hence negative. So its zeros are two complex conjugate numbers given by quadratic formula i.e.
−
2
±
√
2
2
−
4
⋅
1
⋅
3
2
or
−
2
±
√
−
8
2
or
−
1
±
i
√
2
i.e.
−
1
−
i
√
2
and
−
1
+
i
√
2
Now, if
α
and
β
are zeros of quadratic polynomial, then its factors are
(
x
−
α
)
(
x
−
β
)
Hence factors of
x
2
+
2
x
+
3
are
(
x
+
1
+
i
√
2
)
and
(
x
+
1
−
i
√
2
)
and
x
2
+
2
x
+
3
=
(
x
+
1
+
i
√
2
)
(
x
+
1
−
i
√
2
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