Given that \(\frac{2x}{(x + 6)(x + 3)} = \frac{P}{x + 6} + \frac{Q}{x + 3}\),...

Given that \(\frac{2x}{(x + 6)(x + 3)} = \frac{P}{x + 6} + \frac{Q}{x + 3}\), find P and Q.

\(\frac{2x}{(x + 6)(x + 3)} = \frac{P}{x + 6} + \frac{Q}{x + 3}\)

\(\frac{2x}{(x + 6)(x + 3)} = \frac{P(x + 3) + Q(x + 6)}{(x + 6)(x + 3)}\)

Comparing equations, we have

\(2x = Px + 3P + Qx + 6Q\)

\(\implies 3P + 6Q = 0 ... (1) ; P + Q = 2 .... (2)\)

From equation (1), \(3P = -6Q \implies P = -2Q\)

\(\therefore -2Q + Q = -Q = 2 \)

\(Q = -2\)

\(P = -2Q = -2(-2) = 4\)

\(P = 4, Q = -2\)

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