Solve Page 2/(x ^ 2 + 3x + 2) + (5x)/(x ^ 2 - x - 6) + (x + 2)/(x ^ 2 - 2x - 3) | 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

$$\frac { 2 } { x ^ { 2 } + 3 x + 2 } + \frac { 5 x } { x ^ { 2 } - x - 6 } + \frac { x + 2 } { x ^ { 2 } - 2 x - 3 }$$

Evaluate

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

Short Solution Steps

Factor $x^{2}+3x+2$. Factor $x^{2}-x-6$.

$$\frac{2}{\left(x+1\right)\left(x+2\right)}+\frac{5x}{\left(x-3\right)\left(x+2\right)}+\frac{x+2}{x^{2}-2x-3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x+1\right)\left(x+2\right)$ and $\left(x-3\right)\left(x+2\right)$ is $\left(x-3\right)\left(x+1\right)\left(x+2\right)$. Multiply $\frac{2}{\left(x+1\right)\left(x+2\right)}$ times $\frac{x-3}{x-3}$. Multiply $\frac{5x}{\left(x-3\right)\left(x+2\right)}$ times $\frac{x+1}{x+1}$.

$$\frac{2\left(x-3\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)}+\frac{5x\left(x+1\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)}+\frac{x+2}{x^{2}-2x-3}$$

Since $\frac{2\left(x-3\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)}$ and $\frac{5x\left(x+1\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{2\left(x-3\right)+5x\left(x+1\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)}+\frac{x+2}{x^{2}-2x-3}$$

Do the multiplications in $2\left(x-3\right)+5x\left(x+1\right)$.

$$\frac{2x-6+5x^{2}+5x}{\left(x-3\right)\left(x+1\right)\left(x+2\right)}+\frac{x+2}{x^{2}-2x-3}$$

Combine like terms in $2x-6+5x^{2}+5x$.

$$\frac{7x-6+5x^{2}}{\left(x-3\right)\left(x+1\right)\left(x+2\right)}+\frac{x+2}{x^{2}-2x-3}$$

Factor the expressions that are not already factored in $\frac{7x-6+5x^{2}}{\left(x-3\right)\left(x+1\right)\left(x+2\right)}$.

$$\frac{\left(5x-3\right)\left(x+2\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)}+\frac{x+2}{x^{2}-2x-3}$$

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

$$\frac{5x-3}{\left(x-3\right)\left(x+1\right)}+\frac{x+2}{x^{2}-2x-3}$$

Factor $x^{2}-2x-3$.

$$\frac{5x-3}{\left(x-3\right)\left(x+1\right)}+\frac{x+2}{\left(x-3\right)\left(x+1\right)}$$

Since $\frac{5x-3}{\left(x-3\right)\left(x+1\right)}$ and $\frac{x+2}{\left(x-3\right)\left(x+1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{5x-3+x+2}{\left(x-3\right)\left(x+1\right)}$$

Combine like terms in $5x-3+x+2$.

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

Expand $\left(x-3\right)\left(x+1\right)$.

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

Expand

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

Short Solution Steps

Factor $x^{2}+3x+2$. Factor $x^{2}-x-6$.

$$\frac{2}{\left(x+1\right)\left(x+2\right)}+\frac{5x}{\left(x-3\right)\left(x+2\right)}+\frac{x+2}{x^{2}-2x-3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x+1\right)\left(x+2\right)$ and $\left(x-3\right)\left(x+2\right)$ is $\left(x-3\right)\left(x+1\right)\left(x+2\right)$. Multiply $\frac{2}{\left(x+1\right)\left(x+2\right)}$ times $\frac{x-3}{x-3}$. Multiply $\frac{5x}{\left(x-3\right)\left(x+2\right)}$ times $\frac{x+1}{x+1}$.

$$\frac{2\left(x-3\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)}+\frac{5x\left(x+1\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)}+\frac{x+2}{x^{2}-2x-3}$$

Since $\frac{2\left(x-3\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)}$ and $\frac{5x\left(x+1\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{2\left(x-3\right)+5x\left(x+1\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)}+\frac{x+2}{x^{2}-2x-3}$$

Do the multiplications in $2\left(x-3\right)+5x\left(x+1\right)$.

$$\frac{2x-6+5x^{2}+5x}{\left(x-3\right)\left(x+1\right)\left(x+2\right)}+\frac{x+2}{x^{2}-2x-3}$$

Combine like terms in $2x-6+5x^{2}+5x$.

$$\frac{7x-6+5x^{2}}{\left(x-3\right)\left(x+1\right)\left(x+2\right)}+\frac{x+2}{x^{2}-2x-3}$$

Factor the expressions that are not already factored in $\frac{7x-6+5x^{2}}{\left(x-3\right)\left(x+1\right)\left(x+2\right)}$.

$$\frac{\left(5x-3\right)\left(x+2\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)}+\frac{x+2}{x^{2}-2x-3}$$

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

$$\frac{5x-3}{\left(x-3\right)\left(x+1\right)}+\frac{x+2}{x^{2}-2x-3}$$

Factor $x^{2}-2x-3$.

$$\frac{5x-3}{\left(x-3\right)\left(x+1\right)}+\frac{x+2}{\left(x-3\right)\left(x+1\right)}$$

Since $\frac{5x-3}{\left(x-3\right)\left(x+1\right)}$ and $\frac{x+2}{\left(x-3\right)\left(x+1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{5x-3+x+2}{\left(x-3\right)\left(x+1\right)}$$

Combine like terms in $5x-3+x+2$.

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

Expand $\left(x-3\right)\left(x+1\right)$.

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