Solve (x+2)/((x)^(2)-4)+(x-3)/((x)^(2)-9)+(x+3)/((x)^(2)-6x) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{ x+2 }{ { x }^{ 2 } -4 } + \frac{ x-3 }{ { x }^{ 2 } -9 } + \frac{ x+3 }{ { x }^{ 2 } -6x }$$

Evaluate

$\frac{3x^{3}-7x^{2}-9x-18}{x\left(x-6\right)\left(x-2\right)\left(x+3\right)}$

Short Solution Steps

Factor the expressions that are not already factored in $\frac{x+2}{x^{2}-4}$.

$$\frac{x+2}{\left(x-2\right)\left(x+2\right)}+\frac{x-3}{x^{2}-9}+\frac{x+3}{x^{2}-6x}$$

Cancel out $x+2$ in both numerator and denominator.

$$\frac{1}{x-2}+\frac{x-3}{x^{2}-9}+\frac{x+3}{x^{2}-6x}$$

Factor the expressions that are not already factored in $\frac{x-3}{x^{2}-9}$.

$$\frac{1}{x-2}+\frac{x-3}{\left(x-3\right)\left(x+3\right)}+\frac{x+3}{x^{2}-6x}$$

Cancel out $x-3$ in both numerator and denominator.

$$\frac{1}{x-2}+\frac{1}{x+3}+\frac{x+3}{x^{2}-6x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $x-2$ and $x+3$ is $\left(x-2\right)\left(x+3\right)$. Multiply $\frac{1}{x-2}$ times $\frac{x+3}{x+3}$. Multiply $\frac{1}{x+3}$ times $\frac{x-2}{x-2}$.

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

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

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

Combine like terms in $x+3+x-2$.

$$\frac{2x+1}{\left(x-2\right)\left(x+3\right)}+\frac{x+3}{x^{2}-6x}$$

Factor $x^{2}-6x$.

$$\frac{2x+1}{\left(x-2\right)\left(x+3\right)}+\frac{x+3}{x\left(x-6\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-2\right)\left(x+3\right)$ and $x\left(x-6\right)$ is $x\left(x-6\right)\left(x-2\right)\left(x+3\right)$. Multiply $\frac{2x+1}{\left(x-2\right)\left(x+3\right)}$ times $\frac{x\left(x-6\right)}{x\left(x-6\right)}$. Multiply $\frac{x+3}{x\left(x-6\right)}$ times $\frac{\left(x-2\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)}$.

$$\frac{\left(2x+1\right)x\left(x-6\right)}{x\left(x-6\right)\left(x-2\right)\left(x+3\right)}+\frac{\left(x+3\right)\left(x-2\right)\left(x+3\right)}{x\left(x-6\right)\left(x-2\right)\left(x+3\right)}$$

Since $\frac{\left(2x+1\right)x\left(x-6\right)}{x\left(x-6\right)\left(x-2\right)\left(x+3\right)}$ and $\frac{\left(x+3\right)\left(x-2\right)\left(x+3\right)}{x\left(x-6\right)\left(x-2\right)\left(x+3\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(2x+1\right)x\left(x-6\right)+\left(x+3\right)\left(x-2\right)\left(x+3\right)}{x\left(x-6\right)\left(x-2\right)\left(x+3\right)}$$

Do the multiplications in $\left(2x+1\right)x\left(x-6\right)+\left(x+3\right)\left(x-2\right)\left(x+3\right)$.

$$\frac{2x^{3}-12x^{2}+x^{2}-6x+x^{3}+x^{2}-6x+3x^{2}+3x-18}{x\left(x-6\right)\left(x-2\right)\left(x+3\right)}$$

Combine like terms in $2x^{3}-12x^{2}+x^{2}-6x+x^{3}+x^{2}-6x+3x^{2}+3x-18$.

$$\frac{3x^{3}-7x^{2}-9x-18}{x\left(x-6\right)\left(x-2\right)\left(x+3\right)}$$

Expand $x\left(x-6\right)\left(x-2\right)\left(x+3\right)$.

$$\frac{3x^{3}-7x^{2}-9x-18}{x^{4}-5x^{3}-12x^{2}+36x}$$

Expand

$\frac{3x^{3}-7x^{2}-9x-18}{\left(x-2\right)\left(x+3\right)\left(x^{2}-6x\right)}$

Short Solution Steps

Factor the expressions that are not already factored in $\frac{x+2}{x^{2}-4}$.

$$\frac{x+2}{\left(x-2\right)\left(x+2\right)}+\frac{x-3}{x^{2}-9}+\frac{x+3}{x^{2}-6x}$$

Cancel out $x+2$ in both numerator and denominator.

$$\frac{1}{x-2}+\frac{x-3}{x^{2}-9}+\frac{x+3}{x^{2}-6x}$$

Factor the expressions that are not already factored in $\frac{x-3}{x^{2}-9}$.

$$\frac{1}{x-2}+\frac{x-3}{\left(x-3\right)\left(x+3\right)}+\frac{x+3}{x^{2}-6x}$$

Cancel out $x-3$ in both numerator and denominator.

$$\frac{1}{x-2}+\frac{1}{x+3}+\frac{x+3}{x^{2}-6x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $x-2$ and $x+3$ is $\left(x-2\right)\left(x+3\right)$. Multiply $\frac{1}{x-2}$ times $\frac{x+3}{x+3}$. Multiply $\frac{1}{x+3}$ times $\frac{x-2}{x-2}$.

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

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

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

Combine like terms in $x+3+x-2$.

$$\frac{2x+1}{\left(x-2\right)\left(x+3\right)}+\frac{x+3}{x^{2}-6x}$$

Factor $x^{2}-6x$.

$$\frac{2x+1}{\left(x-2\right)\left(x+3\right)}+\frac{x+3}{x\left(x-6\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-2\right)\left(x+3\right)$ and $x\left(x-6\right)$ is $x\left(x-6\right)\left(x-2\right)\left(x+3\right)$. Multiply $\frac{2x+1}{\left(x-2\right)\left(x+3\right)}$ times $\frac{x\left(x-6\right)}{x\left(x-6\right)}$. Multiply $\frac{x+3}{x\left(x-6\right)}$ times $\frac{\left(x-2\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)}$.

$$\frac{\left(2x+1\right)x\left(x-6\right)}{x\left(x-6\right)\left(x-2\right)\left(x+3\right)}+\frac{\left(x+3\right)\left(x-2\right)\left(x+3\right)}{x\left(x-6\right)\left(x-2\right)\left(x+3\right)}$$

Since $\frac{\left(2x+1\right)x\left(x-6\right)}{x\left(x-6\right)\left(x-2\right)\left(x+3\right)}$ and $\frac{\left(x+3\right)\left(x-2\right)\left(x+3\right)}{x\left(x-6\right)\left(x-2\right)\left(x+3\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(2x+1\right)x\left(x-6\right)+\left(x+3\right)\left(x-2\right)\left(x+3\right)}{x\left(x-6\right)\left(x-2\right)\left(x+3\right)}$$

Do the multiplications in $\left(2x+1\right)x\left(x-6\right)+\left(x+3\right)\left(x-2\right)\left(x+3\right)$.

$$\frac{2x^{3}-12x^{2}+x^{2}-6x+x^{3}+x^{2}-6x+3x^{2}+3x-18}{x\left(x-6\right)\left(x-2\right)\left(x+3\right)}$$

Combine like terms in $2x^{3}-12x^{2}+x^{2}-6x+x^{3}+x^{2}-6x+3x^{2}+3x-18$.

$$\frac{3x^{3}-7x^{2}-9x-18}{x\left(x-6\right)\left(x-2\right)\left(x+3\right)}$$

Expand $x\left(x-6\right)\left(x-2\right)\left(x+3\right)$.

$$\frac{3x^{3}-7x^{2}-9x-18}{x^{4}-5x^{3}-12x^{2}+36x}$$