2. The factors of x² - 2x^2 - x + 2 are ..?
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0
Answer:
This quadratic only factors with the help of Complex coefficients:
x
2
−
2
x
+
2
=
(
x
−
1
−
i
)
(
x
−
1
+
i
)
Explanation:
Given:
x
2
−
2
x
+
2
This is in the form
a
x
2
+
b
x
+
c
with
a
=
1
,
b
=
−
2
and
c
=
2
It has discriminant
Δ
given by the formula:
Δ
=
b
2
−
4
a
c
=
(
−
2
)
2
−
4
(
1
)
(
2
)
=
4
−
8
=
−
4
Since
Δ
<
0
, this quadratic has no Real zeros and no linear factors with Real coefficients.
We can still factor it, but we need to use Complex coefficients.
The difference of squares identity can be written:
A
2
−
B
2
=
(
A
−
B
)
(
A
+
B
)
To factor our quadratic, we can complete the square and use the difference of squares identity with
A
=
(
x
−
1
)
and
B
=
i
as follows:
x
2
−
2
x
+
2
=
x
2
−
2
x
+
1
+
1
x
2
−
2
x
+
2
=
(
x
−
1
)
2
+
1
x
2
−
2
x
+
2
=
(
x
−
1
)
2
−
i
2
x
2
−
2
x
+
2
=
(
(
x
−
1
)
−
i
)
(
(
x
−
1
)
+
i
)
x
2
−
2
x
+
2
=
(
x
−
1
−
i
)
(
x
−
1
+
i
)
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