What is the partial fraction decomposition of 7x^2-6x+9/3x(4x^2+9)
The Correct Answer and Explanation is :
The correct answer is: 3×1+4×2+9x−2
To find the partial fraction decomposition of the expression (\frac{7x^2 – 6x + 9}{3x(4x^2 + 9)}), we start by identifying the factors in the denominator, which are (3x) and (4x^2 + 9). The first factor, (3x), is a linear term, and (4x^2 + 9) is a quadratic term that cannot be factored over the reals.
Step 1: Setting Up the Partial Fractions
The general form for the partial fraction decomposition will look like this:
[
\frac{7x^2 – 6x + 9}{3x(4x^2 + 9)} = \frac{A}{3x} + \frac{Bx + C}{4x^2 + 9}
]
Here, (A), (B), and (C) are constants that we need to determine.
Step 2: Combine the Right Side
To combine the right side into a single fraction, we need a common denominator:
[
\frac{A}{3x} + \frac{Bx + C}{4x^2 + 9} = \frac{A(4x^2 + 9) + (Bx + C)(3x)}{3x(4x^2 + 9)}
]
Step 3: Expand and Equate
Expanding the numerator gives:
[
A(4x^2 + 9) + (Bx + C)(3x) = 4Ax^2 + 9A + 3Bx^2 + 3Cx = (4A + 3B)x^2 + 3Cx + 9A
]
Now we equate this to the numerator of the original expression:
[
(4A + 3B)x^2 + 3Cx + 9A = 7x^2 – 6x + 9
]
Step 4: Set Up the System of Equations
From this, we can create a system of equations by comparing coefficients:
- (4A + 3B = 7)
- (3C = -6)
- (9A = 9)
Step 5: Solve for the Constants
From equation 3, we find:
[
A = 1
]
Substituting (A = 1) into equation 1:
[
4(1) + 3B = 7 \implies 3B = 3 \implies B = 1
]
From equation 2:
[
3C = -6 \implies C = -2
]
Step 6: Write the Partial Fraction Decomposition
Now substituting back the values of (A), (B), and (C):
[
\frac{7x^2 – 6x + 9}{3x(4x^2 + 9)} = \frac{1}{3x} + \frac{x – 2}{4x^2 + 9}
]
Conclusion
Thus, the partial fraction decomposition of (\frac{7x^2 – 6x + 9}{3x(4x^2 + 9)}) is:
[
\frac{1}{3x} + \frac{x – 2}{4x^2 + 9}
]
This decomposition allows us to simplify the original expression, making it easier to integrate or analyze in further calculations.