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Express (9x-5)/(2(2x-1)(x-1)) in partial frac

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

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Full solution

Q. Express 9x−52(2x−1)(x−1) \frac{9 x-5}{2(2 x-1)(x-1)} in partial frac
  1. Multiply by Denominator: We want to express (9x−5)/(2(2x−1)(x−1))(9x-5)/(2(2x-1)(x-1)) as A/(2x−1)+B/(x−1)A/(2x-1) + B/(x-1). To find AA and BB, we'll multiply both sides by the denominator 2(2x−1)(x−1)2(2x-1)(x-1).
  2. Equation with A and B: So we have 9x−5=A(2x−1)(x−1)+B(2(2x−1))9x - 5 = A(2x-1)(x-1) + B(2(2x-1)). Now we'll find the values of AA and BB by choosing suitable values for xx.
  3. Plug in x=1x=1: Let's plug in x=1x=1 to get rid of the AA term. We get 9(1)−5=B(2(2(1)−1))9(1) - 5 = B(2(2(1)-1)), which simplifies to 4=2B4 = 2B, so B=2B = 2.
  4. Plug in x=12x=\frac{1}{2}: Now let's plug in x=12x=\frac{1}{2} to get rid of the BB term. We get 9(12)−5=A(2(12)−1)((12)−1)9(\frac{1}{2}) - 5 = A(2(\frac{1}{2})-1)((\frac{1}{2})-1), which simplifies to −4.5=A(0)((−12))-4.5 = A(0)((-\frac{1}{2})), so A=0A = 0.

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