Solve 2x/x^2-9+x/x^2+6x+9-3/x+3

$$\frac{ 2x }{ { x }^{ 2 } -9 } + \frac{ x }{ { x }^{ 2 } +6x+9 } - \frac{ 3 }{ x+3 }$$

Evaluate

$$\frac{3\left(x+9\right)}{\left(x-3\right)\left(x+3\right)^{2}}$$

Short Solution Steps

Factor x^{2}-9. Factor x^{2}+6x+9.

$$\frac{2x}{\left(x-3\right)\left(x+3\right)}+\frac{x}{\left(x+3\right)^{2}}-\frac{3}{x+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+3\right) and \left(x+3\right)^{2} is \left(x-3\right)\left(x+3\right)^{2}. Multiply \frac{2x}{\left(x-3\right)\left(x+3\right)} times \frac{x+3}{x+3}. Multiply \frac{x}{\left(x+3\right)^{2}} times \frac{x-3}{x-3}.

$$\frac{2x\left(x+3\right)}{\left(x-3\right)\left(x+3\right)^{2}}+\frac{x\left(x-3\right)}{\left(x-3\right)\left(x+3\right)^{2}}-\frac{3}{x+3}$$

Since \frac{2x\left(x+3\right)}{\left(x-3\right)\left(x+3\right)^{2}} and \frac{x\left(x-3\right)}{\left(x-3\right)\left(x+3\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{2x\left(x+3\right)+x\left(x-3\right)}{\left(x-3\right)\left(x+3\right)^{2}}-\frac{3}{x+3}$$

Do the multiplications in 2x\left(x+3\right)+x\left(x-3\right).

$$\frac{2x^{2}+6x+x^{2}-3x}{\left(x-3\right)\left(x+3\right)^{2}}-\frac{3}{x+3}$$

Combine like terms in 2x^{2}+6x+x^{2}-3x.

$$\frac{3x^{2}+3x}{\left(x-3\right)\left(x+3\right)^{2}}-\frac{3}{x+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+3\right)^{2} and x+3 is \left(x-3\right)\left(x+3\right)^{2}. Multiply \frac{3}{x+3} times \frac{\left(x-3\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}.

$$\frac{3x^{2}+3x}{\left(x-3\right)\left(x+3\right)^{2}}-\frac{3\left(x-3\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)^{2}}$$

Since \frac{3x^{2}+3x}{\left(x-3\right)\left(x+3\right)^{2}} and \frac{3\left(x-3\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3x^{2}+3x-3\left(x-3\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)^{2}}$$

Do the multiplications in 3x^{2}+3x-3\left(x-3\right)\left(x+3\right).

$$\frac{3x^{2}+3x-3x^{2}-9x+9x+27}{\left(x-3\right)\left(x+3\right)^{2}}$$

Combine like terms in 3x^{2}+3x-3x^{2}-9x+9x+27.

$$\frac{3x+27}{\left(x-3\right)\left(x+3\right)^{2}}$$

Expand \left(x-3\right)\left(x+3\right)^{2}.

$$\frac{3x+27}{x^{3}+3x^{2}-9x-27}$$

Differentiate w.r.t. x

$$-\frac{6\left(\left(x+6\right)^{2}-45\right)}{\left(x-3\right)^{2}\left(x+3\right)^{3}}$$