Solve सरल गनुहास् (Simply ): 1/(x + 2) + 2/(2(x - 2)) - 4/(x ^ 2 - 4); [2U](Ans / (2/(x + 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

$$\frac { 1 } { x + 2 } + \frac { 2 } { 2 ( x - 2 ) } - \frac { 4 } { ( x ^ { 2 } - 4 ) } \quad ( \frac { 2 } { x + 2 }$$

Evaluate

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

Short Solution Steps

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

$$\frac{2\left(x-2\right)}{2\left(x-2\right)\left(x+2\right)}+\frac{2\left(x+2\right)}{2\left(x-2\right)\left(x+2\right)}-\frac{4}{x^{2}-4}\times \frac{2}{x+2}$$

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

$$\frac{2\left(x-2\right)+2\left(x+2\right)}{2\left(x-2\right)\left(x+2\right)}-\frac{4}{x^{2}-4}\times \frac{2}{x+2}$$

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

$$\frac{2x-4+2x+4}{2\left(x-2\right)\left(x+2\right)}-\frac{4}{x^{2}-4}\times \frac{2}{x+2}$$

Combine like terms in $2x-4+2x+4$.

$$\frac{4x}{2\left(x-2\right)\left(x+2\right)}-\frac{4}{x^{2}-4}\times \frac{2}{x+2}$$

Cancel out $2$ in both numerator and denominator.

$$\frac{2x}{\left(x-2\right)\left(x+2\right)}-\frac{4}{x^{2}-4}\times \frac{2}{x+2}$$

Multiply $\frac{4}{x^{2}-4}$ times $\frac{2}{x+2}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{2x}{\left(x-2\right)\left(x+2\right)}-\frac{4\times 2}{\left(x^{2}-4\right)\left(x+2\right)}$$

Multiply $4$ and $2$ to get $8$.

$$\frac{2x}{\left(x-2\right)\left(x+2\right)}-\frac{8}{\left(x^{2}-4\right)\left(x+2\right)}$$

Factor $\left(x^{2}-4\right)\left(x+2\right)$.

$$\frac{2x}{\left(x-2\right)\left(x+2\right)}-\frac{8}{\left(x-2\right)\left(x+2\right)^{2}}$$

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

$$\frac{2x\left(x+2\right)}{\left(x-2\right)\left(x+2\right)^{2}}-\frac{8}{\left(x-2\right)\left(x+2\right)^{2}}$$

Since $\frac{2x\left(x+2\right)}{\left(x-2\right)\left(x+2\right)^{2}}$ and $\frac{8}{\left(x-2\right)\left(x+2\right)^{2}}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x\left(x+2\right)-8}{\left(x-2\right)\left(x+2\right)^{2}}$$

Do the multiplications in $2x\left(x+2\right)-8$.

$$\frac{2x^{2}+4x-8}{\left(x-2\right)\left(x+2\right)^{2}}$$

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

$$\frac{2x^{2}+4x-8}{x^{3}+2x^{2}-4x-8}$$

Expand

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

Short Solution Steps

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

$$\frac{2\left(x-2\right)}{2\left(x-2\right)\left(x+2\right)}+\frac{2\left(x+2\right)}{2\left(x-2\right)\left(x+2\right)}-\frac{4}{x^{2}-4}\times \frac{2}{x+2}$$

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

$$\frac{2\left(x-2\right)+2\left(x+2\right)}{2\left(x-2\right)\left(x+2\right)}-\frac{4}{x^{2}-4}\times \frac{2}{x+2}$$

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

$$\frac{2x-4+2x+4}{2\left(x-2\right)\left(x+2\right)}-\frac{4}{x^{2}-4}\times \frac{2}{x+2}$$

Combine like terms in $2x-4+2x+4$.

$$\frac{4x}{2\left(x-2\right)\left(x+2\right)}-\frac{4}{x^{2}-4}\times \frac{2}{x+2}$$

Cancel out $2$ in both numerator and denominator.

$$\frac{2x}{\left(x-2\right)\left(x+2\right)}-\frac{4}{x^{2}-4}\times \frac{2}{x+2}$$

Multiply $\frac{4}{x^{2}-4}$ times $\frac{2}{x+2}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{2x}{\left(x-2\right)\left(x+2\right)}-\frac{4\times 2}{\left(x^{2}-4\right)\left(x+2\right)}$$

Multiply $4$ and $2$ to get $8$.

$$\frac{2x}{\left(x-2\right)\left(x+2\right)}-\frac{8}{\left(x^{2}-4\right)\left(x+2\right)}$$

Factor $\left(x^{2}-4\right)\left(x+2\right)$.

$$\frac{2x}{\left(x-2\right)\left(x+2\right)}-\frac{8}{\left(x-2\right)\left(x+2\right)^{2}}$$

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

$$\frac{2x\left(x+2\right)}{\left(x-2\right)\left(x+2\right)^{2}}-\frac{8}{\left(x-2\right)\left(x+2\right)^{2}}$$

Since $\frac{2x\left(x+2\right)}{\left(x-2\right)\left(x+2\right)^{2}}$ and $\frac{8}{\left(x-2\right)\left(x+2\right)^{2}}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x\left(x+2\right)-8}{\left(x-2\right)\left(x+2\right)^{2}}$$

Do the multiplications in $2x\left(x+2\right)-8$.

$$\frac{2x^{2}+4x-8}{\left(x-2\right)\left(x+2\right)^{2}}$$

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

$$\frac{2x^{2}+4x-8}{x^{3}+2x^{2}-4x-8}$$