Solve B ( !a 1-a^ 2 - a-2 1+2a )* 2 a+2 + a+1 2-a | 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

$$( \frac { 4 a } { 4 - a ^ { 2 } } - \frac { a - 2 } { 4 + 2 a } ) \cdot \frac { 2 } { a + 2 } + \frac { a + 1 } { 2 - a }$$

Evaluate

$\frac{a+2}{2-a}$

Short Solution Steps

Factor $4-a^{2}$. Factor $4+2a$.

$$\left(\frac{4a}{\left(a-2\right)\left(-a-2\right)}-\frac{a-2}{2\left(a+2\right)}\right)\times \frac{2}{a+2}+\frac{a+1}{2-a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(a-2\right)\left(-a-2\right)$ and $2\left(a+2\right)$ is $2\left(a-2\right)\left(a+2\right)$. Multiply $\frac{4a}{\left(a-2\right)\left(-a-2\right)}$ times $\frac{-2}{-2}$. Multiply $\frac{a-2}{2\left(a+2\right)}$ times $\frac{a-2}{a-2}$.

$$\left(\frac{-2\times 4a}{2\left(a-2\right)\left(a+2\right)}-\frac{\left(a-2\right)\left(a-2\right)}{2\left(a-2\right)\left(a+2\right)}\right)\times \frac{2}{a+2}+\frac{a+1}{2-a}$$

Since $\frac{-2\times 4a}{2\left(a-2\right)\left(a+2\right)}$ and $\frac{\left(a-2\right)\left(a-2\right)}{2\left(a-2\right)\left(a+2\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{-2\times 4a-\left(a-2\right)\left(a-2\right)}{2\left(a-2\right)\left(a+2\right)}\times \frac{2}{a+2}+\frac{a+1}{2-a}$$

Do the multiplications in $-2\times 4a-\left(a-2\right)\left(a-2\right)$.

$$\frac{-8a-a^{2}+2a+2a-4}{2\left(a-2\right)\left(a+2\right)}\times \frac{2}{a+2}+\frac{a+1}{2-a}$$

Combine like terms in $-8a-a^{2}+2a+2a-4$.

$$\frac{-4a-a^{2}-4}{2\left(a-2\right)\left(a+2\right)}\times \frac{2}{a+2}+\frac{a+1}{2-a}$$

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

$$\frac{\left(-a-2\right)\left(a+2\right)}{2\left(a-2\right)\left(a+2\right)}\times \frac{2}{a+2}+\frac{a+1}{2-a}$$

Extract the negative sign in $-2-a$.

$$\frac{-\left(a+2\right)\left(a+2\right)}{2\left(a-2\right)\left(a+2\right)}\times \frac{2}{a+2}+\frac{a+1}{2-a}$$

Cancel out $a+2$ in both numerator and denominator.

$$\frac{-\left(a+2\right)}{2\left(a-2\right)}\times \frac{2}{a+2}+\frac{a+1}{2-a}$$

Multiply $\frac{-\left(a+2\right)}{2\left(a-2\right)}$ times $\frac{2}{a+2}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{-\left(a+2\right)\times 2}{2\left(a-2\right)\left(a+2\right)}+\frac{a+1}{2-a}$$

Cancel out $2\left(a+2\right)$ in both numerator and denominator.

$$\frac{-1}{a-2}+\frac{a+1}{2-a}$$

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

$$\frac{-1}{a-2}+\frac{-\left(a+1\right)}{a-2}$$

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

$$\frac{-1-\left(a+1\right)}{a-2}$$

Do the multiplications in $-1-\left(a+1\right)$.

$$\frac{-1-a-1}{a-2}$$

Combine like terms in $-1-a-1$.

$$\frac{-2-a}{a-2}$$

Expand

$\frac{a+2}{2-a}$

Short Solution Steps

Factor $4-a^{2}$. Factor $4+2a$.

$$\left(\frac{4a}{\left(a-2\right)\left(-a-2\right)}-\frac{a-2}{2\left(a+2\right)}\right)\times \frac{2}{a+2}+\frac{a+1}{2-a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(a-2\right)\left(-a-2\right)$ and $2\left(a+2\right)$ is $2\left(a-2\right)\left(a+2\right)$. Multiply $\frac{4a}{\left(a-2\right)\left(-a-2\right)}$ times $\frac{-2}{-2}$. Multiply $\frac{a-2}{2\left(a+2\right)}$ times $\frac{a-2}{a-2}$.

$$\left(\frac{-2\times 4a}{2\left(a-2\right)\left(a+2\right)}-\frac{\left(a-2\right)\left(a-2\right)}{2\left(a-2\right)\left(a+2\right)}\right)\times \frac{2}{a+2}+\frac{a+1}{2-a}$$

Since $\frac{-2\times 4a}{2\left(a-2\right)\left(a+2\right)}$ and $\frac{\left(a-2\right)\left(a-2\right)}{2\left(a-2\right)\left(a+2\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{-2\times 4a-\left(a-2\right)\left(a-2\right)}{2\left(a-2\right)\left(a+2\right)}\times \frac{2}{a+2}+\frac{a+1}{2-a}$$

Do the multiplications in $-2\times 4a-\left(a-2\right)\left(a-2\right)$.

$$\frac{-8a-a^{2}+2a+2a-4}{2\left(a-2\right)\left(a+2\right)}\times \frac{2}{a+2}+\frac{a+1}{2-a}$$

Combine like terms in $-8a-a^{2}+2a+2a-4$.

$$\frac{-4a-a^{2}-4}{2\left(a-2\right)\left(a+2\right)}\times \frac{2}{a+2}+\frac{a+1}{2-a}$$

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

$$\frac{\left(-a-2\right)\left(a+2\right)}{2\left(a-2\right)\left(a+2\right)}\times \frac{2}{a+2}+\frac{a+1}{2-a}$$

Extract the negative sign in $-2-a$.

$$\frac{-\left(a+2\right)\left(a+2\right)}{2\left(a-2\right)\left(a+2\right)}\times \frac{2}{a+2}+\frac{a+1}{2-a}$$

Cancel out $a+2$ in both numerator and denominator.

$$\frac{-\left(a+2\right)}{2\left(a-2\right)}\times \frac{2}{a+2}+\frac{a+1}{2-a}$$

Multiply $\frac{-\left(a+2\right)}{2\left(a-2\right)}$ times $\frac{2}{a+2}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{-\left(a+2\right)\times 2}{2\left(a-2\right)\left(a+2\right)}+\frac{a+1}{2-a}$$

Cancel out $2\left(a+2\right)$ in both numerator and denominator.

$$\frac{-1}{a-2}+\frac{a+1}{2-a}$$

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

$$\frac{-1}{a-2}+\frac{-\left(a+1\right)}{a-2}$$

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

$$\frac{-1-\left(a+1\right)}{a-2}$$

Do the multiplications in $-1-\left(a+1\right)$.

$$\frac{-1-a-1}{a-2}$$

Combine like terms in $-1-a-1$.

$$\frac{-2-a}{a-2}$$