Given that \(\frac{2x}{(x + 6)(x + 3)} = \frac{P}{x + 6} + \frac{Q}{x + 3}\), find P and Q.
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Correct Answer: Option C
Explanation:
\(\frac{2x}{(x + 6)(x + 3)} = \frac{P}{x + 6} + \frac{Q}{x + 3}\)
\(\frac{2x}{(x + 6)(x + 3)} = \frac{P(x + 3) + Q(x + 6)}{(x + 6)(x + 3)}\)
Comparing equations, we have
\(2x = Px + 3P + Qx + 6Q\)
\(\implies 3P + 6Q = 0 ... (1) ; P + Q = 2 .... (2)\)
From equation (1), \(3P = -6Q \implies P = -2Q\)
\(\therefore -2Q + Q = -Q = 2 \)
\(Q = -2\)
\(P = -2Q = -2(-2) = 4\)
\(P = 4, Q = -2\)
\(\frac{2x}{(x + 6)(x + 3)} = \frac{P}{x + 6} + \frac{Q}{x + 3}\)
\(\frac{2x}{(x + 6)(x + 3)} = \frac{P(x + 3) + Q(x + 6)}{(x + 6)(x + 3)}\)
Comparing equations, we have
\(2x = Px + 3P + Qx + 6Q\)
\(\implies 3P + 6Q = 0 ... (1) ; P + Q = 2 .... (2)\)
From equation (1), \(3P = -6Q \implies P = -2Q\)
\(\therefore -2Q + Q = -Q = 2 \)
\(Q = -2\)
\(P = -2Q = -2(-2) = 4\)
\(P = 4, Q = -2\)