Solve ((x)^(2)+1)/(x)+(x)/((x)^(2)+1)2(1)/(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

$$\frac{ { x }^{ 2 } +1 }{ x } + \frac{ x }{ { x }^{ 2 } +1 } 2 \frac{ 1 }{ 2 }$$

Evaluate

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

Short Solution Steps

Multiply $2$ and $2$ to get $4$.

$$\frac{x^{2}+1}{x}+\frac{x}{x^{2}+1}\times \frac{4+1}{2}$$

Add $4$ and $1$ to get $5$.

$$\frac{x^{2}+1}{x}+\frac{x}{x^{2}+1}\times \frac{5}{2}$$

Multiply $\frac{x}{x^{2}+1}$ times $\frac{5}{2}$ by multiplying numerator times numerator and denominator times denominator.

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

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

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

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

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

Do the multiplications in $\left(x^{2}+1\right)\times 2\left(x^{2}+1\right)+x\times 5x$.

$$\frac{2x^{4}+2x^{2}+2x^{2}+2+5x^{2}}{2x\left(x^{2}+1\right)}$$

Combine like terms in $2x^{4}+2x^{2}+2x^{2}+2+5x^{2}$.

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

Factor the expressions that are not already factored in $\frac{2x^{4}+9x^{2}+2}{2x\left(x^{2}+1\right)}$.

$$\frac{2\left(x^{2}-\left(-\frac{1}{4}\sqrt{65}-\frac{9}{4}\right)\right)\left(x^{2}-\left(\frac{1}{4}\sqrt{65}-\frac{9}{4}\right)\right)}{2x\left(x^{2}+1\right)}$$

Cancel out $2$ in both numerator and denominator.

$$\frac{\left(x^{2}-\left(-\frac{1}{4}\sqrt{65}-\frac{9}{4}\right)\right)\left(x^{2}-\left(\frac{1}{4}\sqrt{65}-\frac{9}{4}\right)\right)}{x\left(x^{2}+1\right)}$$

Expand $x\left(x^{2}+1\right)$.

$$\frac{\left(x^{2}-\left(-\frac{1}{4}\sqrt{65}-\frac{9}{4}\right)\right)\left(x^{2}-\left(\frac{1}{4}\sqrt{65}-\frac{9}{4}\right)\right)}{x^{3}+x}$$

To find the opposite of $-\frac{1}{4}\sqrt{65}-\frac{9}{4}$, find the opposite of each term.

$$\frac{\left(x^{2}+\frac{1}{4}\sqrt{65}+\frac{9}{4}\right)\left(x^{2}-\left(\frac{1}{4}\sqrt{65}-\frac{9}{4}\right)\right)}{x^{3}+x}$$

To find the opposite of $\frac{1}{4}\sqrt{65}-\frac{9}{4}$, find the opposite of each term.

$$\frac{\left(x^{2}+\frac{1}{4}\sqrt{65}+\frac{9}{4}\right)\left(x^{2}-\frac{1}{4}\sqrt{65}+\frac{9}{4}\right)}{x^{3}+x}$$

Use the distributive property to multiply $x^{2}+\frac{1}{4}\sqrt{65}+\frac{9}{4}$ by $x^{2}-\frac{1}{4}\sqrt{65}+\frac{9}{4}$ and combine like terms.

$$\frac{x^{4}+\frac{9}{2}x^{2}-\frac{1}{16}\left(\sqrt{65}\right)^{2}+\frac{81}{16}}{x^{3}+x}$$

The square of $\sqrt{65}$ is $65$.

$$\frac{x^{4}+\frac{9}{2}x^{2}-\frac{1}{16}\times 65+\frac{81}{16}}{x^{3}+x}$$

Multiply $-\frac{1}{16}$ and $65$ to get $-\frac{65}{16}$.

$$\frac{x^{4}+\frac{9}{2}x^{2}-\frac{65}{16}+\frac{81}{16}}{x^{3}+x}$$

Add $-\frac{65}{16}$ and $\frac{81}{16}$ to get $1$.

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

Expand

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

Short Solution Steps

Multiply $2$ and $2$ to get $4$.

$$\frac{x^{2}+1}{x}+\frac{x}{x^{2}+1}\times \frac{4+1}{2}$$

Add $4$ and $1$ to get $5$.

$$\frac{x^{2}+1}{x}+\frac{x}{x^{2}+1}\times \frac{5}{2}$$

Multiply $\frac{x}{x^{2}+1}$ times $\frac{5}{2}$ by multiplying numerator times numerator and denominator times denominator.

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

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

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

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

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

Do the multiplications in $\left(x^{2}+1\right)\times 2\left(x^{2}+1\right)+x\times 5x$.

$$\frac{2x^{4}+2x^{2}+2x^{2}+2+5x^{2}}{2x\left(x^{2}+1\right)}$$

Combine like terms in $2x^{4}+2x^{2}+2x^{2}+2+5x^{2}$.

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

Factor the expressions that are not already factored in $\frac{2x^{4}+9x^{2}+2}{2x\left(x^{2}+1\right)}$.

$$\frac{2\left(x^{2}-\left(-\frac{1}{4}\sqrt{65}-\frac{9}{4}\right)\right)\left(x^{2}-\left(\frac{1}{4}\sqrt{65}-\frac{9}{4}\right)\right)}{2x\left(x^{2}+1\right)}$$

Cancel out $2$ in both numerator and denominator.

$$\frac{\left(x^{2}-\left(-\frac{1}{4}\sqrt{65}-\frac{9}{4}\right)\right)\left(x^{2}-\left(\frac{1}{4}\sqrt{65}-\frac{9}{4}\right)\right)}{x\left(x^{2}+1\right)}$$

Expand $x\left(x^{2}+1\right)$.

$$\frac{\left(x^{2}-\left(-\frac{1}{4}\sqrt{65}-\frac{9}{4}\right)\right)\left(x^{2}-\left(\frac{1}{4}\sqrt{65}-\frac{9}{4}\right)\right)}{x^{3}+x}$$

To find the opposite of $-\frac{1}{4}\sqrt{65}-\frac{9}{4}$, find the opposite of each term.

$$\frac{\left(x^{2}+\frac{1}{4}\sqrt{65}+\frac{9}{4}\right)\left(x^{2}-\left(\frac{1}{4}\sqrt{65}-\frac{9}{4}\right)\right)}{x^{3}+x}$$

To find the opposite of $\frac{1}{4}\sqrt{65}-\frac{9}{4}$, find the opposite of each term.

$$\frac{\left(x^{2}+\frac{1}{4}\sqrt{65}+\frac{9}{4}\right)\left(x^{2}-\frac{1}{4}\sqrt{65}+\frac{9}{4}\right)}{x^{3}+x}$$

Use the distributive property to multiply $x^{2}+\frac{1}{4}\sqrt{65}+\frac{9}{4}$ by $x^{2}-\frac{1}{4}\sqrt{65}+\frac{9}{4}$ and combine like terms.

$$\frac{x^{4}+\frac{9}{2}x^{2}-\frac{1}{16}\left(\sqrt{65}\right)^{2}+\frac{81}{16}}{x^{3}+x}$$

The square of $\sqrt{65}$ is $65$.

$$\frac{x^{4}+\frac{9}{2}x^{2}-\frac{1}{16}\times 65+\frac{81}{16}}{x^{3}+x}$$

Multiply $-\frac{1}{16}$ and $65$ to get $-\frac{65}{16}$.

$$\frac{x^{4}+\frac{9}{2}x^{2}-\frac{65}{16}+\frac{81}{16}}{x^{3}+x}$$

Add $-\frac{65}{16}$ and $\frac{81}{16}$ to get $1$.

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