2x(x+6)(x+3)=Px+6+Qx+3\frac{2x}{(x+6)(x+3)} = \frac{P}{x+6} + \frac{Q}{x+3}(x+6)(x+3)2x=x+6P+x+3Q
2x(x+6)(x+3)=P(x+3)+Q(x+6)(x+6)(x+3)\frac{2x}{(x+6)(x+3)} = \frac{P(x+3)+Q(x+6)}{(x+6)(x+3)}(x+6)(x+3)2x=(x+6)(x+3)P(x+3)+Q(x+6)
Comparing equations, we have
2x=Px+3P+Qx+6Q2x = Px + 3P + Qx + 6Q2x=Px+3P+Qx+6Q
⇒3P+6Q=0…(1);P+Q=2…(2)\Rightarrow 3P + 6Q = 0 \ldots (1) ; P + Q = 2 \ldots (2)⇒3P+6Q=0…(1);P+Q=2…(2)
From equation (1), 3P=−6Q⇒P=−2Q3P = -6Q \Rightarrow P = -2Q3P=−6Q⇒P=−2Q
∴−2Q+Q=−Q=2\therefore -2Q + Q = -Q = 2∴−2Q+Q=−Q=2
Q=−2Q = -2Q=−2
P=−2Q=−2(−2)=4P = -2Q = -2(-2) = 4P=−2Q=−2(−2)=4
P=4,Q=−2P = 4, Q = -2P=4,Q=−2