Factor the following quadratic equations:x4-10x2+9=0
Answers
Answer:
Move
9
to the left side of the equation by adding it to both sides.
x
4
−
10
x
2
+
9
=
0
Rewrite
x
4
as
(
x
2
)
2
.
(
x
2
)
2
−
10
x
2
+
9
=
0
Let
u
=
x
2
. Substitute
u
for all occurrences of
x
2
.
u
2
−
10
u
+
9
=
0
Factor
u
2
−
10
u
+
9
using the AC method.
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(
u
−
9
)
(
u
−
1
)
=
0
Replace all occurrences of
u
with
x
2
.
(
x
2
−
9
)
(
x
2
−
1
)
=
0
Rewrite
9
as
3
2
.
(
x
2
−
3
2
)
(
x
2
−
1
)
=
0
Since both terms are perfect squares, factor using the difference of squares formula,
a
2
−
b
2
=
(
a
+
b
)
(
a
−
b
)
where
a
=
x
and
b
=
3
.
(
x
+
3
)
(
x
−
3
)
(
x
2
−
1
)
=
0
Rewrite
1
as
1
2
.
(
x
+
3
)
(
x
−
3
)
(
x
2
−
1
2
)
=
0
Factor.
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(
x
+
3
)
(
x
−
3
)
(
x
+
1
)
(
x
−
1
)
=
0
If any individual factor on the left side of the equation is equal to
0
, the entire expression will be equal to
0
.
x
+
3
=
0
x
−
3
=
0
x
+
1
=
0
x
−
1
=
0
Set the first factor equal to
0
and solve.
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x
=
−
3
Set the next factor equal to
0
and solve.
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x
=
3
Set the next factor equal to
0
and solve.
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x
=
−
1
Set the next factor equal to
0
and solve.
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x
=
1
The final solution is all the values that make
(
x
+
3
)
(
x
−
3
)
(
x
+
1
)
(
x
−
1
)
=
0
true.
x
=
−
3
,
3
,
−
1
,
1