Solve 6x+9/(x+3)^2=a/(x+3)^2+b/x+3

$$\frac { 6 x + 9 } { ( x + 3 ) ^ { 2 } } = \frac { a } { ( x + 3 ) ^ { 2 } } + \frac { b } { x + 3 }$$

Solve for a

$$a=9-3b+6x-bx$$

$$x\neq -3$$

Solution Steps

Multiply both sides of the equation by \left(x+3\right)^{2}, the least common multiple of \left(x+3\right)^{2},x+3.

$$6x+9=a+\left(x+3\right)b$$

Use the distributive property to multiply x+3 by b.

$$6x+9=a+xb+3b$$

Swap sides so that all variable terms are on the left hand side.

$$a+xb+3b=6x+9$$

Subtract xb from both sides.

$$a+3b=6x+9-xb$$

Subtract 3b from both sides.

$$a=6x+9-xb-3b$$

Solve for b

$$b=-\frac{-6x+a-9}{x+3}$$

$$x\neq -3$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x+3\right)^{2}, the least common multiple of \left(x+3\right)^{2},x+3.

$$6x+9=a+\left(x+3\right)b$$

Use the distributive property to multiply x+3 by b.

$$6x+9=a+xb+3b$$

Swap sides so that all variable terms are on the left hand side.

$$a+xb+3b=6x+9$$

Subtract a from both sides.

$$xb+3b=6x+9-a$$

Combine all terms containing b.

$$\left(x+3\right)b=6x+9-a$$

The equation is in standard form.

$$\left(x+3\right)b=6x-a+9$$

Divide both sides by x+3.

$$\frac{\left(x+3\right)b}{x+3}=\frac{6x-a+9}{x+3}$$

Dividing by x+3 undoes the multiplication by x+3.

$$b=\frac{6x-a+9}{x+3}$$