Solve RE -1031'Ke (3x + 1)/(9x ^ 2 + 3x + 1) + (3x - 1)/(9x ^ 2 - 3x + 1) + 2/(81x ^ 4 + 9x ^ 2 + 1) square ) x ^ 2 - 5x + 6 = | 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

$$\frac { 3 x + 1 } { 9 x ^ { 2 } + 3 x + 1 } + \frac { 3 x - 1 } { 9 x ^ { 2 } - 3 x + 1 } + \frac { 2 } { 81 x ^ { 4 } + 9 x ^ { 2 } + 1 }$$

Evaluate

$\frac{2\left(3x+1\right)}{9x^{2}+3x+1}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $9x^{2}+3x+1$ and $9x^{2}-3x+1$ is $\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)$. Multiply $\frac{3x+1}{9x^{2}+3x+1}$ times $\frac{9x^{2}-3x+1}{9x^{2}-3x+1}$. Multiply $\frac{3x-1}{9x^{2}-3x+1}$ times $\frac{9x^{2}+3x+1}{9x^{2}+3x+1}$.

$$\frac{\left(3x+1\right)\left(9x^{2}-3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{\left(3x-1\right)\left(9x^{2}+3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{2}{81x^{4}+9x^{2}+1}$$

Since $\frac{\left(3x+1\right)\left(9x^{2}-3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}$ and $\frac{\left(3x-1\right)\left(9x^{2}+3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(3x+1\right)\left(9x^{2}-3x+1\right)+\left(3x-1\right)\left(9x^{2}+3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{2}{81x^{4}+9x^{2}+1}$$

Do the multiplications in $\left(3x+1\right)\left(9x^{2}-3x+1\right)+\left(3x-1\right)\left(9x^{2}+3x+1\right)$.

$$\frac{27x^{3}-9x^{2}+3x+9x^{2}-3x+1+27x^{3}+9x^{2}+3x-9x^{2}-3x-1}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{2}{81x^{4}+9x^{2}+1}$$

Combine like terms in $27x^{3}-9x^{2}+3x+9x^{2}-3x+1+27x^{3}+9x^{2}+3x-9x^{2}-3x-1$.

$$\frac{54x^{3}}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{2}{81x^{4}+9x^{2}+1}$$

Factor $81x^{4}+9x^{2}+1$.

$$\frac{54x^{3}}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{2}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}$$

Since $\frac{54x^{3}}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}$ and $\frac{2}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{54x^{3}+2}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}$$

Factor the expressions that are not already factored in $\frac{54x^{3}+2}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}$.

$$\frac{2\left(3x+1\right)\left(9x^{2}-3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}$$

Cancel out $9x^{2}-3x+1$ in both numerator and denominator.

$$\frac{2\left(3x+1\right)}{9x^{2}+3x+1}$$

Use the distributive property to multiply $2$ by $3x+1$.

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

Expand

$\frac{2\left(3x+1\right)}{9x^{2}+3x+1}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $9x^{2}+3x+1$ and $9x^{2}-3x+1$ is $\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)$. Multiply $\frac{3x+1}{9x^{2}+3x+1}$ times $\frac{9x^{2}-3x+1}{9x^{2}-3x+1}$. Multiply $\frac{3x-1}{9x^{2}-3x+1}$ times $\frac{9x^{2}+3x+1}{9x^{2}+3x+1}$.

$$\frac{\left(3x+1\right)\left(9x^{2}-3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{\left(3x-1\right)\left(9x^{2}+3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{2}{81x^{4}+9x^{2}+1}$$

Since $\frac{\left(3x+1\right)\left(9x^{2}-3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}$ and $\frac{\left(3x-1\right)\left(9x^{2}+3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(3x+1\right)\left(9x^{2}-3x+1\right)+\left(3x-1\right)\left(9x^{2}+3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{2}{81x^{4}+9x^{2}+1}$$

Do the multiplications in $\left(3x+1\right)\left(9x^{2}-3x+1\right)+\left(3x-1\right)\left(9x^{2}+3x+1\right)$.

$$\frac{27x^{3}-9x^{2}+3x+9x^{2}-3x+1+27x^{3}+9x^{2}+3x-9x^{2}-3x-1}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{2}{81x^{4}+9x^{2}+1}$$

Combine like terms in $27x^{3}-9x^{2}+3x+9x^{2}-3x+1+27x^{3}+9x^{2}+3x-9x^{2}-3x-1$.

$$\frac{54x^{3}}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{2}{81x^{4}+9x^{2}+1}$$

Factor $81x^{4}+9x^{2}+1$.

$$\frac{54x^{3}}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}+\frac{2}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}$$

Since $\frac{54x^{3}}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}$ and $\frac{2}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{54x^{3}+2}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}$$

Factor the expressions that are not already factored in $\frac{54x^{3}+2}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}$.

$$\frac{2\left(3x+1\right)\left(9x^{2}-3x+1\right)}{\left(9x^{2}-3x+1\right)\left(9x^{2}+3x+1\right)}$$

Cancel out $9x^{2}-3x+1$ in both numerator and denominator.

$$\frac{2\left(3x+1\right)}{9x^{2}+3x+1}$$

Use the distributive property to multiply $2$ by $3x+1$.

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