Solve (1)/(x-2)+(1)/(x+3)+(x+3)/((x-3)^(2)) | 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

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

Evaluate

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

Short Solution Steps

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 $\frac{x-2}{x-2}$.

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

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.

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

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

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

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 $\left(x-3\right)^{2}$ is $\left(x-2\right)\left(x+3\right)\left(x-3\right)^{2}$. Multiply $\frac{2x+1}{\left(x-2\right)\left(x+3\right)}$ times $\frac{\left(x-3\right)^{2}}{\left(x-3\right)^{2}}$. Multiply $\frac{x+3}{\left(x-3\right)^{2}}$ times $\frac{\left(x-2\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)}$.

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

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

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

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

$$\frac{2x^{3}-12x^{2}+18x+x^{2}-6x+9+x^{3}+x^{2}-6x+3x^{2}+3x-18}{\left(x-2\right)\left(x+3\right)\left(x-3\right)^{2}}$$

Combine like terms in $2x^{3}-12x^{2}+18x+x^{2}-6x+9+x^{3}+x^{2}-6x+3x^{2}+3x-18$.

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

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

$$\frac{3x^{3}-7x^{2}+9x-9}{x^{4}-5x^{3}-3x^{2}+45x-54}$$

Expand

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

Short Solution Steps

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 $\frac{x-2}{x-2}$.

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

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.

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

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

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

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 $\left(x-3\right)^{2}$ is $\left(x-2\right)\left(x+3\right)\left(x-3\right)^{2}$. Multiply $\frac{2x+1}{\left(x-2\right)\left(x+3\right)}$ times $\frac{\left(x-3\right)^{2}}{\left(x-3\right)^{2}}$. Multiply $\frac{x+3}{\left(x-3\right)^{2}}$ times $\frac{\left(x-2\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)}$.

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

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

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

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

$$\frac{2x^{3}-12x^{2}+18x+x^{2}-6x+9+x^{3}+x^{2}-6x+3x^{2}+3x-18}{\left(x-2\right)\left(x+3\right)\left(x-3\right)^{2}}$$

Combine like terms in $2x^{3}-12x^{2}+18x+x^{2}-6x+9+x^{3}+x^{2}-6x+3x^{2}+3x-18$.

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

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

$$\frac{3x^{3}-7x^{2}+9x-9}{x^{4}-5x^{3}-3x^{2}+45x-54}$$