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9x + 2 7
[1.1] ------------- + ------------
3x^2 - 2x - 8 3x^2 + x - 4
The quadratic expressions can be factored.
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9x + 2 7
[1.2] --------------- + ---------------
(3x + 4)(x - 2) (3x + 4)(x - 1)
Partial fraction decomposition may now be used on each of the rational expressions.
The first expression:
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A B 9x + 2
[1.3] ------ + ----- = ---------------
3x + 4 x - 2 (3x + 4)(x - 2)
Multiplying by the least common denominator,
This yields the following system of equations:
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[1.5] A + 3B = 9
[1.6] -2A + 4B = 2
or
[1.7] -A + 2B = 1
[1.5] and [1.7] may be added to eliminate A.
By substitution,
So,
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9x + 2 3 2
[1.10] --------------- = ------ + -----
(3x + 4)(x - 1) 3x + 4 x - 2
The second expression:
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A B 7
[1.11] ------ + ----- = ---------------
3x + 4 x - 2 (3x + 4)(x - 1)
Multiplying by the least common denominator,
This yields the following system of equations:
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[1.13] A + 3B = 0
[1.14] -A + 4B = 7
[1.13] and [1.14] may be added to eliminate A.
By substitution,
So,
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7 -3 1
[1.17] --------------- = ------ + -----
(3x + 4)(x - 1) 3x + 4 x - 1
Substituting back into [1.2],
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3 2 3 1
------ + ----- - ------ + -----
3x + 4 x - 2 3x + 4 x - 1
2 1
[1.18] ----- + -----
x - 2 x - 1
What? I like taking the scenic route. Besides, I couldn't pass up a chance to use partial fractions!
Okay, okay. . . Math
is the art of doing as little work as possible. . . .
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9x + 2 7
[2.1] ------------- + ------------
3x^2 - 2x - 8 3x^2 + x - 4
The quadratic expressions can be factored.
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9x + 2 7
[2.2] --------------- + ---------------
(3x + 4)(x - 2) (3x + 4)(x - 1)
The rational expressions may be added together.
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(9x + 2)(x - 1) + 7(x - 2)
--------------------------
(3x + 4)(x - 2)(x - 1)
9x^2 - 16
[2.3] ----------------------
(3x + 4)(x - 2)(x - 1)
The numerator in [2.3] is a difference of squares.
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(3x + 4)(3x - 4)
[2.4] ----------------------
(3x + 4)(x - 2)(x - 1)
[2.4] is simplified by cancelling 3x + 4 in the numerator and denominator.
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3x - 4
[2.5] --------------
(x - 2)(x - 1)
As this is an expression, it can't actually be "solved". However, it is worth noting that the expression is equal to zero when x = 4/3 and undefined when x = 1, x = 2, or x = -4/3. The latter is true because 3x + 4 was in the denominators in the original expression.
P.S.: When you make a post, there is an option to disable smilies. I have mine disabled by default. 8)