Question

What is the partial fraction decomposition of (7x^2 - 6x + 9) / (3x(4x^2 + 9))? Please select the correct partial fraction decomposition from the options provided.

a) 1/(3x)
b) (x - 2)/(4x² + 9)
c) 1/(3x)
d) (x + 2)/(4x² + 9)
e) -2/(3x)
f) (x - 1)/(4x² + 9)
g) -2/(3x)
h) (x + 1)/(4x² + 9)

Answer 1

The partial fraction decomposition of (7x² - 6x + 9) / (3x(4x² + 9)) is (x + 2)/(4x² + 9).

Step-by-step explanation:

The partial fraction decomposition of (7x² - 6x + 9) / (3x(4x² + 9)) is d) (x + 2)/(4x² + 9).

To find the partial fraction decomposition, we first factor the denominator as (4x² + 9) = (2x + 3i)(2x - 3i), where i is the imaginary unit. Then, we write the fraction as A/(3x) + (Bx + C)/(2x + 3i) + (Dx + E)/(2x - 3i), where A, B, C, D, and E are constants to be determined. We then multiply both sides by the denominator and solve for the constants.

In this case, the solution is (x + 2)/(4x² + 9).