Express \(\frac{5x - 12}{(x - 2)(x - 3)}\) in partial fractions
A
\(\frac{2}{x + 2} - \frac{3}{x - 3}\)
B
\(\frac{2}{x - 2} + \frac{3}{x - 3}\)
C
\(\frac{2}{x - 3} - \frac{3}{x - 2}\)
D
\(\frac{5}{x - 3} - \frac{4}{x - 2}\)
Verified Correct Answer: B.
Step-by-Step Mathematical Solution
\(\frac{5x - 12}{(x - 2)(x - 3)} = \frac{A}{x - 2} + \frac{B}{x - 3}\)
= \(\frac{A(x - 3) + B(x - 2)}{(x - 2)(x - 3)}\)
\(\implies 5x - 12 = Ax - 3A + Bx - 2B\)
\(A + B = 5 ... (i)\)
\(-(3A + 2B) = -12 \implies 3A + 2B = 12 ... (ii)\)
From (i), \(A = 5 - B\)
\(3(5 - B) + 2B = 12\)
\(15 - 3B + 2B = 12 \implies B = 3\)
\(A + 3 = 5 \implies A = 2\)
\(\frac{5x - 12}{(x - 2)(x - 3)} = \frac{2}{x - 2} + \frac{3}{x - 3}\)
Source: Myschool Past Questions Archive (Record #30913)