Solve (2a)/(a - 1) - (a ^ 2 + 3)/(a ^ 2 - 1) | 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

$$\frac{2a}{a-1}-\frac{a^{2}+3}{a^{2}-1}$$

Evaluate

$\frac{a+3}{a+1}$

Short Solution Steps

Factor $a^{2}-1$.

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

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

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

Since $\frac{2a\left(a+1\right)}{\left(a-1\right)\left(a+1\right)}$ and $\frac{a^{2}+3}{\left(a-1\right)\left(a+1\right)}$ have the same denominator, subtract them by subtracting their numerators.

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

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

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

Combine like terms in $2a^{2}+2a-a^{2}-3$.

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

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

$$\frac{\left(a-1\right)\left(a+3\right)}{\left(a-1\right)\left(a+1\right)}$$

Cancel out $a-1$ in both numerator and denominator.

$$\frac{a+3}{a+1}$$

Expand

$\frac{a+3}{a+1}$

Short Solution Steps

Factor $a^{2}-1$.

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

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

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

Since $\frac{2a\left(a+1\right)}{\left(a-1\right)\left(a+1\right)}$ and $\frac{a^{2}+3}{\left(a-1\right)\left(a+1\right)}$ have the same denominator, subtract them by subtracting their numerators.

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

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

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

Combine like terms in $2a^{2}+2a-a^{2}-3$.

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

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

$$\frac{\left(a-1\right)\left(a+3\right)}{\left(a-1\right)\left(a+1\right)}$$

Cancel out $a-1$ in both numerator and denominator.

$$\frac{a+3}{a+1}$$