Given that \(\frac{2x}{(x + 6)(x + 3)} = \frac{P}{x + 6} + \frac{Q}{x + 3}\), find P and Q.
Answer Details
We can begin by multiplying both sides of the equation by the denominator \((x+6)(x+3)\). This will eliminate the denominators on the left side of the equation, leaving us with:
$$2x = P(x+3) + Q(x+6)$$
Next, we can expand the right side of the equation:
$$2x = Px + 3P + Qx + 6Q$$
We can then simplify by collecting like terms:
$$(P + Q)x + (3P + 6Q) = 2x$$
Since this equation must be true for all values of x, we can equate the coefficients of x and the constants on both sides of the equation:
$$\begin{aligned} P + Q &= 0 \\ 3P + 6Q &= 2 \end{aligned}$$
Solving the first equation for P in terms of Q, we get:
$$P = -Q$$
Substituting this expression for P into the second equation, we get:
$$3(-Q) + 6Q = 2$$
Simplifying this equation, we get:
$$3Q = 2$$
Thus, we have:
$$Q = \frac{2}{3}$$
Substituting this value for Q into the equation P = -Q, we get:
$$P = -\frac{2}{3}$$
Therefore, we have:
$$P = -\frac{2}{3}\text{ and }Q = \frac{2}{3}$$
So the answer is, P = 4 and Q = -2.