Solve (x + 3)/(x ^ 2 + 3x + 9) + (x - 3)/(x ^ 2 - 3x + 9) - 54/(x ^ 4 + 9x ^ 2 + 81) | 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

$$\frac{x+3}{x^{2}+3x+9}+\frac{x-3}{x^{2}-3x+9}-\frac{54}{x^{4}+9x^{2}+81}$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $x^{2}+3x+9$ and $x^{2}-3x+9$ is $\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)$. Multiply $\frac{x+3}{x^{2}+3x+9}$ times $\frac{x^{2}-3x+9}{x^{2}-3x+9}$. Multiply $\frac{x-3}{x^{2}-3x+9}$ times $\frac{x^{2}+3x+9}{x^{2}+3x+9}$.

$$\frac{\left(x+3\right)\left(x^{2}-3x+9\right)}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}+\frac{\left(x-3\right)\left(x^{2}+3x+9\right)}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}-\frac{54}{x^{4}+9x^{2}+81}$$

Since $\frac{\left(x+3\right)\left(x^{2}-3x+9\right)}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}$ and $\frac{\left(x-3\right)\left(x^{2}+3x+9\right)}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(x+3\right)\left(x^{2}-3x+9\right)+\left(x-3\right)\left(x^{2}+3x+9\right)}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}-\frac{54}{x^{4}+9x^{2}+81}$$

Do the multiplications in $\left(x+3\right)\left(x^{2}-3x+9\right)+\left(x-3\right)\left(x^{2}+3x+9\right)$.

$$\frac{x^{3}-3x^{2}+9x+3x^{2}-9x+27+x^{3}+3x^{2}+9x-3x^{2}-9x-27}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}-\frac{54}{x^{4}+9x^{2}+81}$$

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

$$\frac{2x^{3}}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}-\frac{54}{x^{4}+9x^{2}+81}$$

Factor $x^{4}+9x^{2}+81$.

$$\frac{2x^{3}}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}-\frac{54}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}$$

Since $\frac{2x^{3}}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}$ and $\frac{54}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x^{3}-54}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}$$

Factor the expressions that are not already factored in $\frac{2x^{3}-54}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}$.

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

Cancel out $x^{2}+3x+9$ in both numerator and denominator.

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

Use the distributive property to multiply $2$ by $x-3$.

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $x^{2}+3x+9$ and $x^{2}-3x+9$ is $\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)$. Multiply $\frac{x+3}{x^{2}+3x+9}$ times $\frac{x^{2}-3x+9}{x^{2}-3x+9}$. Multiply $\frac{x-3}{x^{2}-3x+9}$ times $\frac{x^{2}+3x+9}{x^{2}+3x+9}$.

$$\frac{\left(x+3\right)\left(x^{2}-3x+9\right)}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}+\frac{\left(x-3\right)\left(x^{2}+3x+9\right)}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}-\frac{54}{x^{4}+9x^{2}+81}$$

Since $\frac{\left(x+3\right)\left(x^{2}-3x+9\right)}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}$ and $\frac{\left(x-3\right)\left(x^{2}+3x+9\right)}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(x+3\right)\left(x^{2}-3x+9\right)+\left(x-3\right)\left(x^{2}+3x+9\right)}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}-\frac{54}{x^{4}+9x^{2}+81}$$

Do the multiplications in $\left(x+3\right)\left(x^{2}-3x+9\right)+\left(x-3\right)\left(x^{2}+3x+9\right)$.

$$\frac{x^{3}-3x^{2}+9x+3x^{2}-9x+27+x^{3}+3x^{2}+9x-3x^{2}-9x-27}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}-\frac{54}{x^{4}+9x^{2}+81}$$

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

$$\frac{2x^{3}}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}-\frac{54}{x^{4}+9x^{2}+81}$$

Factor $x^{4}+9x^{2}+81$.

$$\frac{2x^{3}}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}-\frac{54}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}$$

Since $\frac{2x^{3}}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}$ and $\frac{54}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x^{3}-54}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}$$

Factor the expressions that are not already factored in $\frac{2x^{3}-54}{\left(x^{2}-3x+9\right)\left(x^{2}+3x+9\right)}$.

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

Cancel out $x^{2}+3x+9$ in both numerator and denominator.

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

Use the distributive property to multiply $2$ by $x-3$.

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