Solve Using t he Quadrat ic Formula
3
1
1
−
x
1
=
+
x + 3
x + 2
x + 1
Move all t erms t o t he left side of t he equat ion and simplify.
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Move all t he expressions t o t he left side of t he equat ion.
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3
1
1
−
x
1
−
x + 3
Simplify
3
−
x + 1
1
1
−
x
= 0
x + 2
−
x + 3
1
−
x + 2
.
x + 1
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Find t he common denominat or.
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3 ((x + 3) (x + 2) (x + 1))
x (x + 1) (x + 2) (x + 3)
x (x + 2) (x + 1)
−
x (x + 1) (x + 2) (x + 3)
x (x + 3) (x + 1)
−
x (x + 1) (x + 2) (x + 3)
x (x + 3) (x + 2)
−
= 0
x (x + 1) (x + 2) (x + 3)
Combine t he numerat ors over t he common denominat or.
3 ((x + 3) (x + 2) (x + 1))
− x (x + 2) (x + 1) − x (x + 3) (x + 1)
− x (x + 3) (x + 2)
= 0
x (x + 1) (x + 2) (x + 3)
Simplify each t erm.
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3x
3
− x
+ 18x
3
− 4x
2
2
+ 33x + 18 − x
− 3x − x
3
− 5x
3
− 3x
2
2
− 2x
− 6x
= 0
x (x + 1) (x + 2) (x + 3)
Subt ract
2x
3
x
from 3x3 .
3
+ 18x
− 4x
2
2
+ 33x + 18 − 3x
− 3x − x
3
− 5x
2
2
− 2x − x
3
− 6x
= 0
x (x + 1) (x + 2) (x + 3)
Subt ract
x
3
x
+ 18x
from 2x3 .
3
2
− 3x − x
+ 33x + 18 − 3x
3
− 5x
2
2
− 2x − 4x
2
− 6x
= 0
x (x + 1) (x + 2) (x + 3)
Combine t he opposit e t erms in
x
3
+ 18x
2
− 3x − x
3
+ 33x + 18 − 3x
− 5x
2
2
− 2x − 4x
2
− 6x
.
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18x
2
− 5x
+ 33x + 18 − 3x
2
2
− 2x − 4x
2
− 3x
− 6x
= 0
x (x + 1) (x + 2) (x + 3)
Subt ract
15x
2
3x
2
2
from 18x .
+ 33x + 18 − 2x − 4x
2
− 3x − 5x
− 6x
2
= 0
x (x + 1) (x + 2) (x + 3)
Subt ract
11x
2
4x
2
from 15x2 .
+ 33x + 18 − 2x − 3x − 5x
2
− 6x
= 0
x (x + 1) (x + 2) (x + 3)
Subt ract
5x
2
from 11x2 .
6x
2
+ 33x + 18 − 2x − 3x − 6x
= 0
x (x + 1) (x + 2) (x + 3)
Subt ract
6x
2
2x
from 33x.
+ 31x + 18 − 3x − 6x
= 0
x (x + 1) (x + 2) (x + 3)
Subt ract
6x
2
3x
from 31x.
+ 28x + 18 − 6x
= 0
x (x + 1) (x + 2) (x + 3)
Subt ract
6x
6x
2
from 28x.
+ 22x + 18
= 0
x (x + 1) (x + 2) (x + 3)
Fact or 2 out of
6x
2
+ 22x + 18
.
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2 (3x
2
+ 11x + 9)
= 0
x (x + 1) (x + 2) (x + 3)
Set t he numerat or equal t o zero.
2 (3x
2
+ 11x + 9) = 0
Solve t he equat ion for x.
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−11 ± √13
x =
6
The result can be shown in mult iple forms.
Exact Form:
−11 ± √13
x =
6
Decimal Form:
x = −1.23240812 … , −2.43425854 …
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