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Express 9x−52(2x−1)(x−1)\frac{9x-5}{2(2x-1)(x-1)} in partial fractions.

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Full solution

Q. Express 9x−52(2x−1)(x−1)\frac{9x-5}{2(2x-1)(x-1)} in partial fractions.
  1. Rewrite and Solve: Rewrite the expression as A2x−1+Bx−1\frac{A}{2x-1} + \frac{B}{x-1} and solve for AA and BB.
  2. Clear Fractions: Multiply both sides by the denominator 2(2x−1)(x−1)2(2x-1)(x-1) to clear the fractions: (9x−5)=A(2(x−1))+B(2x−1)(9x-5) = A(2(x-1)) + B(2x-1).
  3. Choose Values: Choose convenient values for xx to solve for AA and BB. Let x=1x=1, then (9×1−5)=B(2×1−1)(9\times 1-5) = B(2\times 1-1), so 4=B(1)4 = B(1), hence B=4B = 4.
  4. Solve for AA and BB: Now let x=12x=\frac{1}{2}, then (9⋅12−5)=A(2⋅(12−1))(9\cdot\frac{1}{2}-5) = A(2\cdot(\frac{1}{2}-1)), so −4.5=A(2⋅(−12))-4.5 = A(2\cdot(-\frac{1}{2})), hence −4.5=−A-4.5 = -A, so A=4.5A = 4.5.

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