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Answer:
1/(3x) =(x -2)/(4x^2 +9)
Step-by-step explanation:
The denominator cannot be factored further, so the final form will be ...
A/(3x) + (Bx +C)/(4x^2 +9)
The value of this is ...
(A(4x^2 +9) +(3x)(Bx +C))/(3x(4x^2 +9))
= ((3B+4A)x^2 +3Cx +9A)/(3x(4x^2 +9))
Equating coefficients gives ...
3B+4A = 7
3C = -6 ⇒ C = -2
9A = 9 ⇒ A = 1
Using the found value of A, we find ...
3B +4 = 7 ⇒ B = 1
Then the decomposition is ...
[tex]\dfrac{7x^2-6x+9}{3x(4x^2+9)}=\boxed{\dfrac{1}{3x}+\dfrac{x-2}{4x^2+9}}[/tex]
